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7 Lyapunov-Razumikhin techniques for state-dependent delay

differential equations

A.R. Humphriesa, F.M.G. Magpantayb

aDepartment of Mathematics and Statistics, McGill University, 805 Sherbrooke St. W., Montreal, QC, Canada H3A 0B9bDepartment of Mathematics, University of Manitoba, 186 Dysart Road, Winnipeg, MB, Canada R3T 2N2

Abstract

We present Lyapunov stability and asymptotic stability theorems for steady state solutions of

general state-dependent delay differential equations (DDEs) using Lyapunov-Razumikhin meth-

ods. Our results apply to DDEs with multiple discrete state-dependent delays, which may be

nonautonomous for the Lyapunov stability result, but autonomous (or periodically forced) for

the asymptotic stability result. Our main technique is to replace the DDE by a nonautonomous

ordinary differential equation (ODE) where the delayed terms become source terms in the ODE.

The asymptotic stability result and its proof are entirely new, and based on a contradiction ar-

gument together with the Arzela-Ascoli theorem. This approach alleviates the need to construct

auxiliary functions to ensure the asymptotic contraction, which is a feature of all other Lyapunov-

Razumikhin asymptotic stability results of which we are aware.

We apply our results to a state-dependent model equation which includes Hayes equation as a

special case, to directly establish asymptotic stability in parts of the stability domain along with

lower bounds on the size of the basin of attraction.

Keywords: delay differential equations, asymptotic stability, Lyapunov-Razumikhin theorem

1. Introduction

We consider the following general delay differential equation (DDE) in d dimensions with N

discrete state-dependent delays,

{

u(t) = f(

t, u(t), u(t − τ1(t, u(t))), . . . , u(t − τN(t, u(t))))

, t > t0,

u(t) = ϕ(t), t 6 t0,(1.1)

and prove Lyapunov stability and asymptotic stability results using Lyapunov-Razumikhin tech-

niques. We apply our results to the model state-dependent DDE

{

u(t) = µu(t) + σu(t − a − cu(t)), t > 0,

u(t) = ϕ(t), t 6 0,(1.2)

Email addresses: tony.humphries@mcgill.ca (A.R. Humphries), felicia.magpantay@umanitoba.ca

(F.M.G. Magpantay)

Preprint submitted to arxiv March 29, 2017

with a > 0, as an example of (1.1), to directly show asymptotic stability in parts of the stability

domain, and derive bounds on the basin of attraction.

Differential equations with state-dependent delays arise in many applications including milling

[21], control theory [41], haematopoiesis [6] and economics [29]. There is a well-established

theory for retarded functional differential equations (RFDEs) as infinite-dimensional dynamical

systems on function spaces [15, 7, 10], which encompasses problems with constant or prescribed

delay, but very little of this theory is directly applicable to state-dependent delay problems. Ex-

tending the theory to state-dependent DDEs, including equations of the form (1.1) is the subject

of ongoing study. See [16] for further examples and a review of recent progress.

The model state-dependent DDE (1.2) includes the constant delay DDE often known as Hayes

equation [17] as a special case when c = 0. Hayes equation is a standard model problem used to

illustrate stability theory for constant delay DDEs in most texts on the subject including [15, 39,

20], as well as being a standard numerical analysis test problem [2]. Hayes equation is also used

to illustrate Lyapunov-Razumikhin stability results in [1, 15, 37].

The state-dependent DDE (1.2) was introduced by Mallet-Paret and Nussbaum, and is a natural

generalisation of Hayes equation to a state-dependent DDE with a single delay which is linearly

state-dependent. But whereas Hayes equation is linear, the DDE (1.2) is nonlinear and can admit

bounded periodic solutions. Mallet-Paret and Nussbaum investigate the existence and form of

the slowly oscillating periodic solutions of a singularly perturbed version of (1.2) in detail in [35]

and use it as an illustrative example for more general problems in [32, 33, 34]. This DDE ia also

studied in [3, 18, 19, 24, 31].

Following [15], to define an RFDE let Rd be the d-dimensional linear vector space over the

real numbers equipped with the Euclidean inner product · and Euclidean norm | · |. Let r > 0 and

C = C(

[−r, 0],Rd)

be the Banach space of continuous functions mapping [−r, 0] to Rd with the

supremum norm denoted ‖ · ‖. If u ∈ C(

[t0 − r, t f ],Rd)

then for every t ∈ [t0, t f ] define ut ∈ C by

ut(θ) = u(t + θ), θ ∈ [−r, 0]. (1.3)

Then for F : R ×C → Rd, with the dot denoting a right-derivative, an RFDE is defined by

u(t) = F(t, ut), ut0 = ϕ ∈ C. (1.4)

A solution of the RFDE (1.4) is a function u ∈ C(

[t0 − r, t f ],Rd)

which satisfies (1.4) for t ∈[t0 − r, t f ). If F is Lipschitz on R ×C then existence and uniqueness of solutions is assured [15],

and the RFDE (1.4) defines a dynamical system with the function space C as its phase space.

If a constant function ϕ ∈ C yields F(t, ϕ) = 0 for all t then ϕ is a steady state of the RFDE. To

study the stability of steady states of RFDEs, the method of Lyapunov functions for ODEs was

first extended to Lyapunov functionals V : R×C → R for RFDEs by Krasovskii [25]. Lyapunov

theorems for stability of RFDEs require the time derivative of the functional along a solution of

(1.4) to be nonpositive or strictly negative, similar to the theorems for ODEs using Lyapunov

functions [15, 25]. However, finding functionals V : R×C → R with this property for RFDEs is

much harder than in the ODE case where the Lyapunov functions have the form V : R×Rd → R.

Razumikhin [38] developed the theory on how one might go from using the more difficult

Lyapunov functionals for RFDEs back to Lyapunov functions again. His fundamental idea is

that it is only necessary to require a constraint on the derivative of V whenever the solution is

about to exit a ball centered at the steady state. Following this approach, Barnea [1] presents

a Lyapunov stability theorem for RFDEs and also considers Hayes equation. A comprehensive

discussion of Lyapunov functionals and functions for general RFDEs is presented by Hale and

2

Verduyn Lunel in chapter 5 of [15]. Other works with Razumikhin-type results include [14, 22,

23, 25, 26, 27, 37, 43]. Of these [27, 37, 43] include results tailored for time-dependent delays.

State-dependent DDEs of the form (1.1) can easily be written as RFDEs. For example, with

F(t, ϕ) = µϕ(0) + σϕ(−a − cϕ(0)), (1.5)

the model problem (1.2) is an RFDE of the form (1.4). However, two problems arise. Firstly, we

need an a priori bound r on the delays in (1.1) to consider it as an RFDE. This can be overcome

for specific problems such as (1.2) for which bounds can be derived under certain parameter

conditions. The second difficulty is more fundamental; it is well known that state-dependency of

the delays results in F not being Lipschitz on R ×C [16]. This is easily shown directly for (1.5).

The Lyapunov-Razumikhin results of Barnea [1] only establish Lyapunov stability and assume

that F is Lipschitz and so are not applicable to state-dependent DDEs of the form (1.1) when they

are written as RFDEs. Other authors, such as Hale and Verduyn Lunel [15], make the weaker

assumptions on F, but use auxiliary functions to establish Lyapunov stability and uniform asymp-

totic stability. The construction of these functions is nontrivial in all but the simplest examples,

and we have not seen such functions constructed for a state-dependent problem. Rather than try

to circumvent these problems for RFDEs, in Section 2 we will develop new proofs of Lyapunov

stability and asymptotic stability for the state-dependent DDE (1.1) with N discrete delays.

In Section 2 we present our main Lyapunov-Razumikhin stability results. In Assumption 2.1

we state the assumptions that we make on the nonautonomous DDE (1.1) with N (state-

dependent) delays, the main ones being that f is locally Lipschitz with respect to its arguments in

Rd and the delays are locally bounded near to the steady state. Then in Theorem 2.5 we provide

sufficient conditions for Lyapunov stability of a steady state of the DDE. The main idea behind

the proof is the conversion of the DDE into an auxiliary ODE problem where the delayed terms

are regarded as source terms. In Theorem 2.7 we establish asymptotic stability of the steady state

when the DDE (1.1) is autonomous. For simplicity of exposition we present the proof for the

case of a single delay, but the result remains true for multiple delays or periodically forced prob-

lems. This result is significantly different to previous Lyapunov-Razumikhin asymptotic stability

results which require auxiliary functions to establish uniform asymptotic stability. In contrast,

Theorem 2.7 does not require the construction of any auxiliary functions, and is proved using the

auxiliary ODE by a contradiction argument, which shows there cannot exist a solution which is

not asymptotic to the steady state.

Theorems 2.5 and 2.7 establish Lyapunov stability and asymptotic stability when the solutions

of the auxiliary ODE have certain properties, but to determine those properties exactly would

require the solutions of the DDE. So, in Section 2, we also show how to define a family of

ODE problems that are subject to constraints defined by bounds on the DDE solution and its

derivatives, which can be determined without solving the DDE. Lemma 2.3 establishes bounds on

the growth of solutions to (1.1) which is used to ensure solutions remain bounded for sufficiently

long (k times the largest delay for some integer k) to acquire k bounded derivatives. Stability is

then established from the solution properties of this family of constrained ODE problems.

In Section 3 we review the stability region of the model equation (1.2) which is known [12] to

be the same for the state-dependent (c , 0) and constant delay cases (c = 0). We also consider

the properties of the auxiliary ODE we define for this problem, and define sets and functions that

are required in the following sections to apply our Lyapunov-Razumikhin results.

In Section 4 we apply Theorem 2.7 to provide a proof of asymptotic stability of the steady state

of model equation (1.2) in subsets of the known stability region, together with lower bounds on

3

the size of the basin of attraction. This result is given as Theorem 4.7. Since the delayed inputs

to the auxiliary ODE have k − 1 bounded derivatives, for the k = 2 and k = 3 results we establish

suitable bounds on the first and second derivatives of the ODE source terms, while the k = 1

result, does not require any differentiability of these terms.

The expressions for the stability regions derived in Sections 4 all involve a term that needs to be

maximized over a closed interval (see the definitions of the P(δ, c, k) functions in Definition 3.5).

For k = 1 this maximum is readily evaluated, while for k = 2 an expression for the maximum

is established in Theorem 4.2 (whose proof is given in Appendix A). Plots and measurements of

the derived asymptotic stability regions in (µ, σ) parameter space are given in Section 5, where it

is seen that these regions grow with the integer k, but do not appear to fill out the entire stability

region in the case µ , 0. In Section 5, we also briefly review previous work on the c = 0 constant

delay case of (1.2) with µ = 0 and µ , 0 and point out an error in the results of Barnea [1].

In Section 6 we present two examples of solutions which are not asymptotic to the steady state

of the model equation (1.2) when µ > 0 > σ, and which hence give upper bounds on the radius

of the largest ball contained in the basin of attraction of the steady state. We compare these with

the lower bounds on the basin of attraction given by Theorem 4.7 for k = 1, 2 and 3. In Section 7

we present brief conclusions, and compare and contrast our approach with linearization.

2. Lyapunov-Razumikhin techniques for state-dependent DDEs

Here we state and prove our main theorems to establish the Lyapunov stability and asymptotic

stability of steady state solutions to state-dependent DDEs of the form (1.1).

We will consider continuous initial functions ϕ ∈ C, where for the state-dependent DDE (1.1)

we let C = C([−r(δ), 0],Rd), with r(δ) defined by (2.1) below. By a solution of (1.1) we mean a

function u ∈ C1([t0, t f ),Rd) which satisfies (1.1) for t ∈ [t0, t f ) for t f > t0 and u(t0) = ϕ(t0). We

denote by B(0, δ) the closed ball centred at zero with radius δ in Rd. We will assume that (1.1)

has a steady state at u = 0 and make the following assumptions throughout.

Assumption 2.1. Let d, N and k ∈ Z, d > 1, N > 1, k > 1 and t0 ∈ R.

1. f : R × R(N+1)d → Rd and τi : [t0,∞) × Rd → R for i = 1, . . . ,N are continuous functions

of their variables.

2. f (t, 0, 0, . . . , 0) = 0 for all t > t0.

3. τi(t, 0) > 0 for all t > t0 and i = 1, ...,N and the constant τmax = maxi=1,...,N supt>t0τi(t, 0)

satisfies τmax ∈ (0,∞).

4. There exist positive constants L0, L1, . . . , LN and δ0 such that∣

∣

∣ f (t, u, v1, . . . , vN) − f (t, u, v1, . . . , vN)∣

∣

∣ 6 L0|u − u| + L1|v1 − v1| + · · · + LN |vN − vN |

for all t > t0 and u, v1, . . . , vN , u, v1, . . . , vN ∈ B(0, δ0) ⊂ Rd. Let L = L0 + L1 + · · · + LN .

5. The delay terms τi(t, u) are nonnegative and Lipschitz continuous in u on [t0,∞) × B(0, δ0)

with Lipschitz constants Lτi.

6. f (t, u, v1, . . . , vN) is at least max{k − 2, 0} times differentiable in its u and v variables, and

τi(t, u) is at least max{k − 1, 0} times differentiable in u.

Definition 2.2. For any δ ∈ (0, δ0] define

r(δ) = supt>t0,|u|6δ,

i=1,...,N

τi(t, u). (2.1)

4

Assumption 2.1, items 3 and 5 imply that 0 < τmax 6 r(δ) 6 τmax + δmaxi Lτi< ∞ with

limδ→0 r(δ) = τmax > 0. In contrast, the bound on the delay terms used in RFDE theory by Hale

and Verduyn Lunel [15] is a global bound on τi(t, u) over all t and u.

The Lipschitz continuity conditions in Assumption 2.1, items 4–5 are based on those of Driver

[8]. From the results in [8], these conditions ensure local existence and uniqueness of a solution

to (1.1) close to the steady state solution for all sufficiently small Lipschitz continuous initial

functions ϕ, and existence of a solution for all sufficiently small continuous initial functions ϕ.

We will prove Lyapunov stability and asymptotic stability by contradiction, by showing that

there is no solution of (1.1) that does not have the required stability property. Thus we do not

need to make assumptions specifically to ensure existence or uniqueness of solutions. Rather, we

make sufficient assumptions so solutions that do exist have the properties we require.

Since with state-dependent delays it can be hard ensure a priori that the delays τi(t, u(t)) are

strictly positive along solutions, and problems with vanishing delays can be interesting, we do

not assume that delays τi(t, u) are strictly positive in Assumption 2.1. This makes our results

more widely applicable, allowing us to deal with state-dependent DDEs without having to first

impose or prove that the delays do not vanish along solutions. The possibility of vanishing delays

will on the other hand complicate some of our proofs, starting with Lemma 2.3 below. Similar

results to Lemma 2.3 are well known for time-varying and nonvanishing delays and proved using

a Gronwall lemma and the method of steps; however the method of steps is not applicable with

vanishing delays, and we will need a more technical proof.

The following lemma provides bounds on the growth of solutions to (1.1). For initial functions

ϕ ∈ C we will use Part II of Lemma 2.3 to ensure that solutions u(t) remain bounded for a

sufficiently long time interval so that u(t) acquires the differentiability that we require. Part I of

Lemma 2.3 will be essential in the proof of asymptotic stability in Theorem 2.7.

Lemma 2.3. Suppose that Assumption 2.1 is satisfied.

I. Let T > t0 and ε > 0. Suppose that u(t) and u(t) are solutions to (1.1) with initial functions

ϕ and ϕ ∈ C respectively, where both |ϕ(t)| 6 δ0 and |ϕ(t)| 6 δ0 for t 6 t0, and |u(t)| 6 δ0

and |u(t)| 6 δ0 for all t ∈ [t0, T ]. If also |ϕ(t) − ϕ(t)| 6 ε for t 6 t0 then

|u(t) − u(t)| 6 εeL(t−t0) ∀t ∈ [t0, T ].

II. Let T > t0 and δ ∈ (0, δ0]. Let |ϕ(t)| 6 δeL(t0−T ) for t 6 t0. Then any solution of (1.1)

defined for t ∈ [t0, T ] satisfies

|u(t)| 6 δeL(t−T )6 δ ∀t ∈ [t0, T ].

Proof. This proof is similar to the derivation given in Halanay [13] for RFDEs of the form (1.4),

and which Halanay credits to Krasovskii. We prove part I first. For L > L suppose there exists

t∗ ∈ [t0, T ) such that Part I holds with L replaced by L only up to t = t∗. That is

|u(t) − u(t)| 6 εeL(t−t0)

for all t ∈ [t0, t∗], but that for every t > t∗ there exists t ∈ (t∗, t] such that |u(t)−u(t)| > εeL(t−t0). The

simplest situation would be where for some h > 0 we have |u(t)−u(t)| > εeL(t−t0) for t ∈ (t∗, t∗+h),

but the existence of such an interval is neither guaranteed by the conditions of the lemma, nor

needed in this proof. First we note that

|u(t)− u(t)| ddt|u(t)− u(t)| = 1

2ddt|u(t)− u(t)|2 = [u(t)− u(t)] · [u(t)− ˙u(t)] 6 |u(t)− u(t)| |u(t) − ˙u(t)|.

5

Since |u(t∗) − u(t∗)| = εeL(t∗−t0) > 0, dividing by the common factor leads to

|u(t∗) − ˙u(t∗)| > ddt|u(t) − u(t)|

∣

∣

∣

t=t∗. (2.2)

Since the solutions to (1.1) have right derivatives for t > t0, we can also compute

ddt|u(t) − u(t)|

∣

∣

∣

t=t∗= lim

t→t+∗

|u(t) − u(t)| − |u(t∗) − u(t∗)|t − t∗

> limt→t+∗

eL(t−t0) − eL(t∗−t0)

t − t∗

ε = LεeL(t∗−t0).

Since t∗−τi(t∗, u(t∗)) 6 t∗ it follows that |u(t∗−τi(t∗, u(t∗)))− u(t∗−τi(t∗, u(t∗)))| 6 εeL(t∗−t0). But,

using Assumption 2.1, item 4 we see that

|u(t∗) − ˙u(t∗)| =∣

∣

∣ f(

t∗, u(t∗), u(t∗ − τ1(t∗, u(t∗))), . . . , u(t∗ − τN(t∗, u(t∗))))

− f(

t∗, u(t∗), u(t∗ − τ1(t∗, u(t∗))), . . . , u(t∗ − τN(t∗, u(t∗))))

∣

∣

∣,

6 L0|u(t∗) − u(t∗)| + L1|u(t∗ − τ1(t∗, u(t∗))) − u(t∗ − τ1(t∗, u(t∗)))|. . . + LN |u(t∗ − τN(t∗, u(t∗))) − u(t∗ − τN(t∗, u(t∗)))|,

6 (L0 + L1 + . . . + LN)εeL(t∗−t0) = LεeL(t∗−t0) < LεeL(t∗−t0).

Thus |u(t∗) − ˙u(t∗)| < ddt|u(t) − u(t)|

∣

∣

∣

t=t∗, but this contradicts (2.2), hence there does not exist such

a t∗, and so |u(t) − u(t)| 6 εeL(t−t0) for all t ∈ [t0, T ]. Since this holds for all L > L, part I follows.

We prove part II by contradiction. If part II is false then for some initial function ϕ such that

|ϕ(t)| 6 δeL(t0−T ) for t 6 t0 there exists T ∗ ∈ (t0, T ) and a solution u(t) such that |u(T ∗)| > δeL(T ∗−T )

but |u(t)| 6 δ0 for all t ∈ [t0, T∗]. Letting ϕ(t) = 0 for all t < 0 from Assumption 2.1, item 2 we

also have the solution u(t) = 0 for all t > t0. Now applying part I on the interval [t0, T∗] with

these initial conditions and ε = δeL(t0−T ) we find that

|u(T ∗)| = |u(T ∗) − u(T ∗)| 6 δeL(t0−T )eL(T ∗−t0) = δeL(T ∗−T ),

which contradicts |u(T ∗)| > δeL(T ∗−T ) and so part II of the lemma holds.

The growth bounds on the solution in Lemma 2.3 are valid even in the case of vanishing delays.

If τi(t, u) were bounded away from zero, a Gronwall argument could be used to obtain a tighter

bound, but we allow for the possibility of vanishing delays in the state-dependent case.

In this section we will develop a constructive technique for showing Lyapunov stability and

asymptotic stability based on Lyapunov-Razumikhin ideas. To establish Lyapunov stability in

Theorem 2.5 we will show that solutions remain in a closed ball B(0, δ) of radius δ about the

steady state. We are thus implicitly using the Lyapunov function V(u) = |u|2/2, though it will not

appear directly in our results. Asymptotic stability is established in Theorem 2.7, by showing

that all solutions that remain in B(0, δ) must converge to the steady state. The application of these

results to the model problem (1.2) is demonstrated in Sections 3–4.

We will prove Lyapunov stability (and later asymptotic stability) by contradiction. Suppose

the DDE (1.1) has a solution which escapes the ball B(0, δ) at time t, then u(t) = x with |x| = δ.Now letting v(θ) = u(t + θ) and

ηi(θ) = u(t + θ − τi(t + θ, u(t + θ))), θ ∈ [−r(δ), 0], i = 1, . . . ,N

for this solution u(t), we can rewrite the DDE (1.1) as a nonautonomous ODE

v(θ) = f (t + θ, v(θ), η1(θ), . . . , ηN(θ))

6

where

v(−τi(t, x)) = u(t − τi(t, x)) = ηi(0), i = 1, . . . ,N.

Hence for each i such that τi(t, x) > 0 the escaping solution of the DDE (1.1) corresponds a

solution of an ODE boundary value problem (BVP) with v(−τi(t, x)) = ηi(0) and v(0) = x. To

establish stability it is sufficient to show that the ODE BVP does not have any such solutions. We

will use Lemma 2.3 part II to ensure that |u(t)| 6 δ for t 6 t0 + kr(δ) so that the forcing functions

ηi(θ) acquire the regularity we require. Hence we define the set of forcing functions ηi which

could correspond to an escaping trajectory as follows.

Definition 2.4. Suppose that Assumption 2.1 is satisfied for (1.1) and k > 1. Let δ ∈ (0, δ0],

|x| = δ and t > t0 + kr(δ). Define the set,

E(k)(δ, x, t) =

{

(η1, . . . , ηN) : ηi ∈ Ck−1(

[−r(δ), 0], B(0, δ))

,

and conditions 1 and 2 are satisfied.

}

. (2.3)

1. x · f (t, x, η1(0), . . . , ηN(0)) > 0,

2. For some initial function ϕ ∈ C, equation (1.1) has solution u(t) such that ηi(θ) = u(t + θ −τi(t, u(t + θ)) for θ ∈ [−r(δ), 0] for each i ∈ {1, . . . ,N}.

Condition (1) in the definition is equivalent to ddt|u(t)| > 0, a necessary condition for the

solution to escape the ball B(0, δ) at time t. In the following theorem we prove the first of our

main results, that if certain conditions hold for all functions in the sets E(k)(δ, x, t) then the steady

state of (2.9) is Lyapunov stable. The condition (2.5) implies that the ODE BVP discussed above

has no solutions, while (2.6) allows solutions with ddθ|v(θ)|θ=0 6 0.

The sets E(k)(δ, x, t) however cannot be determined without solving (1.1), so it is not practical

to actually solve for them. Instead, the conditions of the theorems can be shown to hold for larger

sets that contain E(k)(δ, x, t). We prove the theorems first and then consider such larger sets.

Theorem 2.5. Suppose that Assumption 2.1 is satisfied for (1.1). For δ ∈ (0, δ0], x ∈ Rd, |x| = δ,define E(k)(δ, x, t) as in Definition 2.4. Consider the family of auxiliary ODE problems,

{

v(θ) = f (t + θ, v(θ), η1(θ), . . . , ηN(θ)), θ ∈ [−τi(t, x), 0],

v(−τi(t, x)) = ηi(0) ∈ Rd,(2.4)

for i = 1, . . . ,N and t > t0 + kr(δ). We denote the solution of (2.4) by v(x, η1, . . . , ηN)(θ) if

we want to emphasize the dependence on x and ηi, or just v(θ) otherwise. Suppose there exists

δ1 ∈ (0, δ0] such that for all δ ∈ (0, δ1), and for every x such that |x| = δ and all t > t0 + kr(δ),

for all (η1, . . . , ηN) ∈ E(k)(δ, x, t) the solution of (2.4) for some I ∈ {1, . . . ,N} satisfies τI (t, x) > 0

and either1δv(x, η1, . . . , ηN)(0) · x < δ, (2.5)

or1δv(x, η1, . . . , ηN)(0) · x = δ, and v(x, η1, . . . , ηN)(0) · x 6 0, (2.6)

then the zero solution to (1.1) is Lyapunov stable. Moreover if δ ∈ (0, δ1) and |ϕ(s)| < δe−Lkr(δ)

for s ∈ [t0 − r(δ), t0] then the solution of (1.1) satisfies |u(t)| 6 δ for all t > t0.

Proof. Let the hypothesis of the theorem hold and let δ ∈ (0, δ1). For λ ∈ (0, 1) suppose |ϕ(s)| 6λδe−Lkr(δ) for s ∈ [t0−r(δ), t0]. By Lemma 2.3 part II, the solution of (1.1) satisfies u(t) ∈ B(0, λδ)

for all t 6 t0 + kr(δ). We will prove that u(t) ∈ B(0, δ) for all t > t0 by contradiction.

7

t0

δ

−δ

δe−Lkr

−δe−Lkr

δeL(t−t0−kr)

−δeL(t−t0−kr)

ϕ(t)

t0 + kr t∗ t∗∗ t t

|u(t∗)| = δ

|u(t∗∗)| = δ∗∗

|u(t)| = u

|u(t)| = δ

Figure 1: Illustration of the proof of Theorem 2.5 in one dimension.

Assume that a solution escapes the ball B(0, δ) for the first time at some time t∗ > t0 + kr(δ),

then there exists δ ∈ (δ, δ1) and t ∈ (t∗,∞) such that |u(t)| = δ > δ. But |u(t∗)| = δ so by the

continuity of u(t), for each δ ∈ (δ, δ) there exists t ∈ (t∗, t) such that |u(t)| = δ.It follows from Assumption 2.1 and equation (2.1) that r(δ) is a continuous function of δ and

hence δe−Lkr(δ) is a continuous function of δ as well. Thus we can choose δ ∈ (δ, δ1) sufficiently

small so that λδe−Lkr(δ) < δe−Lkr(δ) for all δ ∈ (δ, δ).

For such a δ, noting that u(t) ∈ Ck+1 ⊆ C1 for t > t∗, and |u(t)| > |u(t∗)| = δ, there must exist

t∗∗ ∈ (t∗, t) such that ddt|u(t∗∗)| > 0 and |u(t∗∗)| = δ∗∗ = supt∈[t∗ ,t∗∗] |u(t)| ∈ (δ, δ). This solution

escapes the ball B(0, δ∗∗) for the first time at t∗∗ > t∗ > t0 + kr(δ). Let x = u(t∗∗) so |x| = δ∗∗ and

0 < 12

ddt|u(t∗∗)|2 = u(t∗∗) · u(t∗∗),

= u(t∗∗) · f(

t∗∗, u(t∗∗), u(t∗∗ − τ1(t∗∗, u(t∗∗))), . . . , u(t∗∗ − τN(t∗∗, u(t∗∗))))

,

= x · f(

t∗∗, x, u(t∗∗ − τ1(t∗∗, u(t∗∗))), . . . , u(t∗∗ − τN(t∗∗, u(t∗∗))))

. (2.7)

Since |ϕ(s)| 6 λδe−Lkr(δ) < δ∗∗e−Lkr(δ∗∗), it follows from Lemma 2.3 part II that t∗∗ > t0+kr(δ∗∗).Now consider the auxiliary ODE problem (2.4) for I ∈ {1, . . . ,N} such that either (2.5) or (2.6)

holds. Let v(θ) = u(t∗∗ + θ), and noting that [−τI(t∗∗, x), 0] ⊆ [−r(δ∗∗), 0], let ηi(θ) = u

(

t∗∗ + θ −τi(t∗∗ + θ, u(t∗∗ + θ))

)

for θ ∈ [−r(δ∗∗), 0]. Then

t∗∗ + θ − τi(t∗∗ + θ, u(t∗∗ + θ)) > t0 + kr(δ∗∗) − 2r(δ∗∗) = t0 + (k − 2)r(δ∗∗), for θ ∈ [−r(δ∗∗), 0].

It follows that ηi ∈ Ck−1([−r(δ∗∗), 0], B(0, δ∗∗)) for i = 1, . . . ,N. From (2.7) we deduce that

x · f (t∗∗, x, η1(0), . . . , ηN(0)) > 0. (2.8)

Furthermore, we can consider

ηi(θ) = U(t0 + kr(δ∗∗) + θ − τi(t0 + kr(δ∗∗) + θ,U(t0 + kr(δ∗∗) + θ)), for θ ∈ [−r(δ∗∗), 0],

where U is the solution to (1.1) with initial function Φ defined as

Φ(s) =

{

ϕ(s − kr(δ∗∗) + t∗∗), s 6 t0 + kr(δ∗∗) − t∗∗,u(s − kr(δ∗∗) + t∗∗), s ∈ [t0 + kr(δ∗∗) − t∗∗, t0].

8

Thus, (η1, . . . , ηN) ∈ E(k)(δ∗∗, x, t∗∗). Moreover,

v(θ) = u(t∗∗ + θ),

= f(

t∗∗ + θ, u(t∗∗ + θ), u(t∗∗ + θ − τ1(t∗∗ + θ, u(t∗∗ + θ))), . . . , u(t∗∗ + θ − τN (t∗∗ + θ, u(t∗∗ + θ))))

,

= f(

t∗∗ + θ, v(θ), η1(θ), . . . , ηN(θ))

.

Also,

v(−τI(t∗∗, x)) = u

(

t∗∗ − τI(t∗∗, x)

)

= u(

t∗∗ − τI(t∗∗, u(t∗∗))

)

= ηI(0).

Thus v is a solution to the ODE system (2.4) with i = I and (η1, . . . , ηN) ∈ E(k)(δ∗∗, x, t∗∗). But

v(0) = u(t∗∗) = x with |x| = δ∗∗ so 1δ∗∗ v(x, η1, . . . , ηN)(0) · x = δ∗∗ which contradicts (2.5).

Meanwhile, equation (2.8) implies that v(x, η1, . . . , ηN)(0) · x > 0 which contradicts (2.6). Thus

any solution which escapes the ball B(0, δ∗∗) violates the conditions of the theorem, so u(t) ∈B(0, δ∗∗) ⊆ B(0, δ) for all t > t0. But this is true for all δ ∈ (δ, δ) and hence if |ϕ(s)| 6 λδe−Lkr(δ)

then |u(t0, ϕ)(t)| 6 δ for all t > t0, establishing Lyapunov stability. Since this holds for all

λ ∈ (0, 1) the result follows.

The proof of Theorem 2.5 is complicated by the auxiliary ODE (2.4) being nonautonomous.

The solution of a nonautonomous ODE escaping the ball B(0, δ) for the first time at t∗ neither

implies that ddt|u(t∗)| > 0 nor that there exists t∗∗ > t∗ such that |u(t)| > δ for all t ∈ (t∗, t∗∗). As

an illustration of this, consider the function y(t) = δ + t3 sin2(2π/t) which is easily seen to be

continuously differentiable and crosses δ at t = 0 with y′(0) = 0, and for which there does not

exist any ε > 0 such that y(t) > δ for all t ∈ (0, ε).

Our second main result is to show asymptotic stability of the steady state u = 0 if the auxiliary

ODE (2.4) satisfies the strict inequality (2.6). We will do this for autonomous DDEs, and for

simplicity of notation we only present the derivation for problems with one delay term (N = 1).

The extension to multiple delays is straightforward, and we discuss the extension to periodically

forced nonautonomous DDEs after Theorem 2.7. Hence we consider autonomous DDEs of the

form (1.1) with N = 1 for which f (t, u, v) = f (0, u, v) and τ1(t, u) = τ(0, u) for all t. In this case

we may set t0 = 0 and rewrite (1.1) as{

u(t) = f(

0, u(t), u(t − τ(0, u(t))))

, t > 0,

u(t) = ϕ(t) ∈ C, t 6 0,(2.9)

where C = C([−r(δ), 0],Rd). By Assumption 2.1 item 2 the DDE (2.9) has the trivial steady

state solution u = 0. We write the solution to (2.9) as u(ϕ)(t) when we want to emphasize

initial conditions, or just u(t) otherwise. For equation (2.9) the auxiliary ODE introduced in (2.4)

becomes{

v(θ) = f(

0, v(θ), η(θ))

, θ ∈ [−τ(0, x), 0],

v(−τ(0, x)) = η(0),(2.10)

and since we consider a single delay, there is one such auxiliary ODE associated with (2.9). We

write the solution of (2.10) as v(x, η)(θ) if we want to emphasize the dependence on x and η,

or just v(θ) otherwise. The sets E(k)(δ, x, t) defined in (2.3) are no longer dependent on t for an

autonomous DDE, and so we denote them by E(k)(δ, x) which for (2.9) is defined as follows.

Definition 2.6. Suppose that Assumption 2.1 is satisfied for (2.9) and k > 1. Let δ ∈ (0, δ0] and

|x| = δ. Define the set

E(k)(δ, x) =

η : η ∈ Ck−1([−τ(0, x), 0], B(0, δ))

, such that x · f (0, x, η(0)) > 0,

and for some initial function ϕ ∈ C the solution u(t) of (2.9) satisfies

η(θ) = u(kr(δ) + θ − τ(0, u(kr(δ) + θ)) for θ ∈ [−τ(0, x), 0]

. (2.11)

9

To show asymptotic stability of the zero solution to (2.9) it is sufficient to strengthen the

conditions of Theorem 2.5 by requiring that solutions of the auxiliary ODE problem satisfy the

strict inequality (2.5) and not the weaker condition (2.6). For the DDE (2.9) the condition (2.5)

becomes1

δv(x, η)(0) · x < δ, (2.12)

and we now show asymptotic stability when all solutions of the auxiliary ODE satisfy (2.12).

Theorem 2.7 (Asymptotic stability). Suppose that Assumption 2.1 is satisfied for (2.9). For

δ ∈ (0, δ0], x ∈ Rd, |x| = δ, define E(k)(δ, x) as in Definition 2.6. If there exists δ1 ∈ (0, δ0] such

that for all δ ∈ (0, δ1), and for every x such that |x| = δ, for all η ∈ E(k)(δ, x) the solution v(x, η)(θ)

of the auxiliary ODE problem (2.10) satisfies (2.12) then the results of Theorem 2.5 hold and

moreover, the zero solution of (2.9) is asymptotically stable. Furthermore, if |ϕ(s)| < δ1e−Lkr(δ1)

for s ∈ [−r(δ1), 0] then u(t)→ 0 as t → ∞.

Proof. The only differences between the conditions of Theorem 2.5 and Theorem 2.7 is that

Theorem 2.5 allows a finite number of delays and nonautonomous f and requires the solution of

the auxiliary ODE problem (2.4) to satisfy (2.5) or (2.6), while Theorem 2.7 assumes autonomous

f , one delay, and requires that the strict inequality (2.12) hold. Thus it trivially follows that the

requirements of Theorem 2.5 are satisfied, and the results of Theorem 2.5 hold.

Let δ ∈ (0, δ1), r = r(δ) and |ϕ(s)| 6 δe−Lkr for s ∈ [−r, 0]. Then by Theorem 2.5 we have

|u(t)| 6 δ for all t > 0. Consider such a solution. Since |u(t)| 6 δ for all t > 0 we have

lim supt→∞ |u(t)| = δ∞ with δ∞ ∈ [0, δ], and it remains only to show that δ∞ = 0.

Since lim supt→∞ |u(t)| = δ∞ and {u : |u| = δ∞} is compact in Rd there exists ti such that

limi→∞ ti = ∞ and limi→∞ u(ti) = x∞ with |x∞| = δ∞. Assume without loss of generality that

ti > (k + 1)r for all i.

Since |u(t)| 6 δ for all t > 0, and |ϕ(t)| 6 δe−Lkr 6 δ for t 6 0 it follows from Assumption 2.1

items 2 and 4 that | ddt

u(t)| 6 (L0 + L1)δ for all t > 0.

Now, consider the sequence of functions vi(θ) = u(ti+θ) for θ ∈ [−(k+1)r, 0]. These functions

and their derivatives are uniformly bounded with ‖vi‖ 6 δ and ‖ ddt

vi‖ 6 (L0 + L1)δ. The set of

all C1 functions satisfying these bounds forms a unifromly bounded and equicontinuous closed

family of functions defined on compact set [−(k + 1)r, 0]. By the Arzela-Ascoli theorem the

sequence of functions vi(θ) has a uniformly convergent subsequence. Let {vi} now denote this

subsequence and let v(θ) be the limiting function, which has v(0) = x∞. Note that |v(θ)| 6 δ∞for all θ ∈ [−(k + 1)r, 0], since the existence of a point with |v(θ)| > δ∞ would contradict that

lim supt→∞ |u(t)| = δ∞.

Let ϕ∗(θ) = v(−kr + θ) for θ ∈ [−r, 0] then we claim that the solution of (2.9) with initial

function ϕ∗ is u∗(t) = v(t−kr) for t ∈ [0, kr]. To see that this is true, let supt∈[0,kr] |u∗(t)−v(t−kr)| =ε > 0. Now let ui(t) solve (2.9) with corresponding initial functions ϕi(θ) = vi(−kr + θ) for

θ ∈ [−r, 0], so ui(t) = vi(t− kr) for t ∈ [0, kr]. For all i sufficiently large we have supt∈[0,kr] |ui(t)−v(t − kr)| = supt∈[0,kr] |vi(t − kr) − v(t − kr)| 6 1

3ε by the uniform convergence of the vi to

v. But also by the uniform convergence for all i sufficiently large we have |ϕi(θ) − ϕ∗(θ)| =|vi(−kr + θ) − v(−kr + θ)| 6 1

3εe−Lkr for all θ ∈ [−r, 0], and hence by Lemma 2.3 part I we have

supt∈[0,kr] |ui(t) − u∗(t)| = supt∈[0,kr] |vi(t − kr) − u∗(t)| 6 13ε. But now

ε = supt∈[0,kr]

|u∗(t) − v(t − kr)| 6 supt∈[0,kr]

|vi(t − kr) − u∗(t)| + supt∈[0,kr]

|vi(t − kr) − v(t − kr)| = 23ε,

10

which can only be true if ε = 0 so the solution of (2.9) with ϕ∗(θ) = v(−kr + θ) for θ ∈ [−r, 0] is

indeed u∗(t) = v(t − kr) for t ∈ [0, kr].

Now let η(θ) = v(θ − τ(0, v(θ))) for θ ∈ [−τ(0, x∞), 0] which implies that η(θ) = u∗(kr + θ −τ(0, u∗(kr + θ))). Moreover |v(θ)| 6 δ∞ for all θ ∈ [−(k + 1)r, 0] implies that |η(θ)| 6 δ∞ for

θ ∈ [−τ(0, x∞), 0] and hence η ∈ Ck−1(

[−τ(0, x∞), 0], B(0, δ∞))

. To show that η ∈ E(k)(δ∞, x∞) it

remains only to show that x∞ · f (0, x∞, η(0)) > 0. But if this is false then

0 > x∞ · f (0, x∞, η(0)) = v(0) · f (0, v(0), η(0)) = u∗(kr) · f (0, u∗(kr), u∗(kr − τ(0, u∗(kr))))

= u∗(kr) · u∗(kr) = 12

ddt|u∗(kr)|.

But, |u∗(kr)| = δ∞ and ddt|u∗(kr)| < 0 implies that there exists ε > 0 such that |u∗(t)| > δ∞ for

t ∈ (kr − ε, kr), or equivalently |v(t)| > δ∞ for t ∈ (−ε, 0). But this contradicts |v(θ)| 6 δ∞ for all

θ ∈ [−(k + 1)r, 0], so we must have x∞ · f (0, x∞, η(0)) > 0 and η ∈ E(k)(δ∞, x∞).

Now v(0) = x∞ implies v(0) · x∞ = δ2∞. But unless δ∞ = 0 this contradicts that (2.12) holds for

all δ ∈ (0, δ1). The result follows.

Notice that Theorem 2.7 not only establishes asymptotic stability of the steady state, but also

shows that the basin of attraction of the steady state contains the ball

{

ϕ : ‖ϕ‖ < δ1e−Lkr(δ1)}. (2.13)

We will consider the basin of attraction of the steady state of the model problem (1.2) in Section 6.

The extension of Theorem 2.7 to multiple delays is straightforward. The proof given above

would not be valid for nonautonomous DDEs. However the proof would only fail in one crucial

step; for a general nonautonomous DDE (1.1), the limiting function v(t) would not in general de-

fine a solution of the DDE. The result is easily extended to periodically nonautonomous DDEs by

choosing the initial sequence ti to be ti = (k+1)r+iT where T is the period of the nonautonomous

function f , and if necessary taking a subsequence so that u(ti) converges to x∞.

Our asymptotic stability result and its proof differs very significantly from other asymptotic

stability results for RFDEs which are all similar to Theorem 4.2 of Hale and Verduyn Lunel

[15]. Beyond the technical differences in continuity assumptions, and whether delays are locally

or globally bounded, there are two fundamental but related differences between our result and

results such as those in [15]. Firstly, in Theorem 2.7 we establish asymptotic stability, but in The-

orem 4.2 of [15] the stronger property of uniform asymptotic stability is obtained. But secondly,

auxiliary functions with specific properties are required (in Theorem 4.2 of [15] four auxiliary

functions, u, v, ω and p appear) to obtain the contraction that leads to the uniform asymptotic

stability. Construction of such functions is difficult even for constant delay DDEs, and a major

obstacle to the application of these theorems. In contrast, we use a proof by contradiction which

shows that there does not exist a trajectory which is not asymptotic to the steady state. The con-

tradiction argument establishes asymptotic stability rather than uniform asymptotic stability, but

does not require any troublesome auxiliary functions, and thus is much easier to apply. In the

following sections we will use Theorem 2.7 to study the asymptotic stability of the steady state

of the model state-dependent DDE (1.2).

We next define the larger sets containing E(k)(δ, x) in which we will later show that conditions

of Theorem 2.7 hold to establish asymptotic stability for the model problem (1.2). By items 4–6

in Assumption 2.1, if a bound on u(t) is given for t ∈ [−r(δ), (k− 1)r(δ)] we can also find bounds

on up to the k−1 order derivatives of u(

t− τ(0, u(t))

for t ∈ [(k−1)r(δ), kr(δ)]. These bounds can

11

be derived from the bounds on f , τ and their derivatives. Recalling the definition of E(k)(δ, x) in

Definition 2.6 this leads us to the following definition.

Definition 2.8. Suppose that Assumption 2.1 is satisfied for (2.9) and k > 1. Let δ ∈ (0, δ0]

while |x| = δ. Let the functionsD j(δ) be Lipschitz continuous in δ for j = 0, . . . , k−1 and satisfy

D j(δ) > supt∈[(k−1)r(δ),kr(δ)]

∣

∣

∣

d j

dt j u(

t − τ(t, u(t)))

∣

∣

∣, (2.14)

given that |u(t)| 6 δ for all t ∈ [−r(δ), kr(δ)], where u(t) is a solution to (2.9). Define the set

E(k)(δ, x) =

{

η : η ∈ PCk−1(

[−τ(0, x), 0], B(0, δ))

, x · f (0, x, η(0)) > 0,∣

∣

∣

d j

dθ j η(θ)∣

∣

∣ 6 D j(δ) for θ ∈ [−τ(0, x), 0], j = 0, . . . , k − 1

}

(2.15)

where PCk−1([−τ(0, x), 0], B(0, δ))

denotes the space of Ck−2 functions which are piecewise Ck−1.

Clearly, E(k)(δ, x) ⊆ E(k)(δ, x). It is convenient to consider piecewise Ck−1 functions in Defini-

tion 2.8 because we will later seek the supremum of an integral over the set E(k)(δ, x). Even if all

the functions in E(k)(δ, x) were Ck−1, in general the maximiser could still be piecewise Ck−1.

In Section 4 we derive bounds D j(δ) for the model problem (1.2), and use these to identify

parameter regions for which all η ∈ E(k)(δ, x) satisfy (2.12), and hence the steady state of (1.2) is

asymptotically stable by Theorem 2.7. For δ ∈ (0, δ0], x ∈ Rd and |x| = δ, it is useful to define

G(δ, x) = supη∈E(k)(δ,x)

1

δv(x, η)(0) · x, F (δ) = sup

|x|=δG(δ, x), (2.16)

where v(x, η) is the solution to (2.10). Notice that for δ ∈ (0, δ0] and |x| = δ we have

supη∈E(k) (δ,x)

1

δv(x, η)(0) · x 6 sup

η∈E(k)(δ,x)

1

δv(x, η)(0) · x = G(δ, x) 6 sup

|x|=δG(δ, x) = F (δ). (2.17)

Thus if F (δ) < δ for all δ ∈ (0, δ1) then (2.12) holds for all δ ∈ (0, δ1) and Theorems 2.5 and 2.7

can be applied. Although F (δ) < δ is a somewhat stronger condition than (2.12) we will find it

convenient to work with when considering the model problem (1.2).

The set E(k)(δ, x, t) given by (2.3) for the DDE (1.1) can be easily generalised to a larger set

E(k)(δ, x, t), in a similar manner. For t > t0 + kr(δ) we let

E(k)(δ, x, t) =

(η1, . . . , ηN) : ηi ∈ PCk−1(

[−r(δ), 0], B(0, δ))

,

x · f (t, x, η1(0), . . . , ηN(0)) > 0,∣

∣

∣

d j

dθ j ηi(θ)∣

∣

∣ 6 Di j(δ, t) for θ ∈ [−r(δ), 0], i = 1, . . . ,N, j = 0, . . . , k − 1

(2.18)

where for all solutions u to (1.1) which satisfy |u(s)| 6 δ for s ∈ [t − (k + 1)r(δ), t],

Di j(δ, t) > sups∈[t−r(δ),t]

∣

∣

∣

∣

d j

dt j u(

s − τi(s, u(s)))

∣

∣

∣

∣. (2.19)

It follows that E(k)(δ, x, t) ⊆ E(k)(δ, x, t), and hence establishing properties on the set E(k)(δ, x, t)

is sufficient to apply Theorem 2.5. However, we will consider the autonomous model problem

(1.2) in the following sections, and so will not need to consider E(k)(δ, x, t) or E(k)(δ, x, t) further.

12

3. Model Equation Properties

In the following sections we will apply the Lyapunov-Razumikhin theory of Section 2 to the

model state-dependent DDE given in (1.2). In this section we consider the properties of the DDE

(1.2) and its auxiliary ODE (2.10), and will define the sets and functions that we will use to

apply our results to this model problem. We begin by considering boundedness and, existence

and uniqueness of solutions of the DDE (1.2) with µ + σ < 0, which generalise the results of

Mallet-Paret and Nussbaum in [35] for σ < µ < 0.

Lemma 3.1. With c , 0, let µ + σ < 0 < a and suppose u ∈ C1([0,∞),R) solves (1.2) for t > 0

with cϕ(0) > −a then t − a − cu(t) < t for all t > 0.

Proof. The model DDE (1.2) is invariant under the transformation u 7→ −u, c 7→ −c, so we

consider only the case c > 0. Suppose ϕ(0) > −a/c and let t∗ > 0 be the first time for which

u(t∗) = −a/c. Then u(t) > −a/c for t < t∗ implies u(t∗) 6 0, but from (1.2) with u(t∗) = −a/c we

have u(t∗) = (µ + σ)u(t∗) = − ac(µ + σ) > 0, supplying the required contradiction. If ϕ(0) = −a/c

then u(0) > 0 and the result follows similarly.

We will always consider the DDE (1.2) with a > 0 and µ+σ < 0, then Lemma 3.1 assures that

the deviating argument is always a delay. The lemma also gives the lower bound u(t) > −a/c on

solutions when c > 0 (or an upper bound on solutions when c < 0). When µ < 0 we can bound

solutions above and below. It is convenient to define

M0 = −a

c, N0 =

aσ

cµ, τ = a + cN, τ0 = a + cN0. (3.1)

We will use [M0,N0] and also [M,N] as bounds on solutions of the single delay DDE (1.2) (in

contrast to the multiple delay DDE (1.1) for which we used N to denote the number of delays).

Lemma 3.2. Let µ + σ < 0 < a and µ < 0 and suppose u ∈ C1([0,∞),R) solves (1.2) for

t > 0. If σ > 0 let sign(c)M ∈ sign(c)[M0, 0) and sign(c)N > 0, and suppose that sign(c)ϕ(t) ∈sign(c)[M,N] for all t ∈ [−τ, 0]. If σ 6 0 let M = M0 and N = max{N0, ϕ(0)} and suppose

sign(c)ϕ(t) > sign(c)M0 for all t ∈ [−τ, 0]. Then

sign(c)u(t) ∈ sign(c)(M,N), ∀t > 0. (3.2)

Proof. Again, we consider the c > 0 case, then it is sufficient to show that u(t) > 0 if u(t) = M,

and u(t) < 0 if u(t) = N given that u(s) ∈ (M,N) for s ∈ (0, t). The case where u(t) = M0 is dealt

with in the proof of Lemma 3.1, the other cases are straightforward.

Theorem 3.3. Let µ + σ < 0 < a. Let the initial history function ϕ be continuous and for

µ < 0 satisfy the bounds given in Lemma 3.2. For µ > 0 let sign(c)ϕ(t) > sign(c)M0 for all

t ∈ (−∞, 0]. Then there exists at least one solution u ∈ C1([0,∞),R) which solves (1.2) for all

t > 0. If µ < 0 any solution satisfies the bounds (3.2), while if µ > 0 any solution satisfies

sign(c)u(t) > sign(c)M0 for all t > 0. If ϕ is locally Lipschitz the solution is unique.

Proof. Local existence and uniqueness follows directly from the results of Driver [8], and for

µ < 0 global existence and uniqueness follows from the extended existence result of Driver [8]

using the bounds on the delay and solution given by Lemma 3.1 and 3.2. The only delicate case

13

delay-dependent

delay-independent

µ

σ

Σ∆

Σ∆

Σc

Σw

Figure 2: The analytic stability region Σ⋆ in the (µ,σ) plane, divided into the delay-independent cone Σ∆, and the

delay-dependent wedge Σw and cusp Σc.

is for −σ > µ > 0 for which (considering the case c > 0) Lemma 3.1 gives only a lower bound,

u(t) > M0. But then u(t) 6 µu(t) + σM0 and the Gronwall lemma implies that

u(t) 6(

ϕ(0) + σµ

M0

)

eµt − σµ

M0 = (ϕ(0) − N0)eµt + N0. (3.3)

Since ϕ(0) > M0 > N0 in this case, solutions cannot become unbounded in finite time, and

global existence again follows. For this case ϕ(t) should be defined for all t 6 0 since with the

exponentially growing bound (3.3) on u(t) it is possible that t − a− cu(t)→ −∞ as t → +∞.

As already mentioned in the introduction, the constant delay DDE known as Hayes equation,

which corresponds to (1.2) with c = 0 has been much studied. The (µ, σ) values for which its

steady state is asymptotically stable when a > 0 and c = 0 are well known (see eg. [15]) and

given in Definition 3.4.

Definition 3.4 (Stability region Σ⋆). Let a > 0 and c = 0. Let Σ⋆ be the open set of the (µ, σ)-

parameter space between the curves

ℓ⋆ ={

(s,−s) : s ∈ (−∞, 1/a]}

, g⋆ ={

(µ(s), σ(s)) : s ∈ (0, π/a)}

where the functions µ(s) and σ(s) are given by

µ(s) = s cot(as), σ(s) = −s csc(as). (3.4)

The stability region Σ⋆ is further divided into three subregions: the cone Σ∆ = {(µ, σ) : |σ| < −µ},the wedge Σw = (Σ⋆ \Σ∆)∩{µ < 0} and the cusp Σc = Σ⋆∩{µ > 0}, which are shown in Figure 2.

Σ⋆ is the parameter region in the (µ, σ)-plane for which the zero solution to the DDE (1.2) is

locally asymptotically stable in both the constant and state-dependent delay cases. The cone Σ∆forms the delay-independent stability region (because this does not change when a is changed)

while Σw ∪Σc is often referred to as the delay-dependent stability region. For the constant delay

14

case (c = 0) this region is found from the characteristic equation [9]. The results of Gyori

and Hartung [12] show the state-dependent case (c , 0) of (1.2) has the same (exponentially)

asymptotic stability region. On the boundary of Σ⋆ the steady-state is Lyapunov stable for the

constant delay case, and the stability is delicate in the state-dependent case [40].

In this paper we derive new proofs of stability in parts of Σ⋆ for the state-dependent case

using Theorem 2.7. The asymptotic stability of the zero solution to (1.2) in all of the delay-

independent region ((µ, σ) ∈ Σ∆) will be shown in Theorem 4.1. In Theorem 4.7 we will also

show asymptotic stability of the steady state of the model problem (1.2) for (µ, σ) in subsets of

Σw ∪Σc, by applying Theorem 2.7 with k = 1 to 3. Here we define some notation that will be

required. Let (µ, σ) ∈ Σw ∪Σc, k ∈ Z, k > 1, δ0 ∈ (0, |a/c|) and δ ∈ (0, δ0). It is easy to see that

Assumption 2.1 is satisfied for (1.2) with L0 = |µ|, L1 = |σ|, τmax = a and r(δ) = a + |c|δ. Thus

for the model problem (1.2) the sets E(k)(δ, x) from Definition 2.6 are given by

E(k)(δ, x) =

η : η ∈ Ck−1([−a − cx, 0], [−δ, δ]), µx2 + σxη(0) > 0,

and for some initial function ϕ ∈ C equation (2.9) has solution

η(θ) = u(kr(δ) + θ − a − cu(kr(δ) + θ)) for θ ∈ [−a − cx, 0]

. (3.5)

To apply the stability theorems in the next section we will derive boundsD j(δ) for j = 0, . . . , k−1

and δ ∈ (0, δ0] as in Definition 2.8. Once these bounds are determined, the sets E(k)(δ, x) from

Definition 2.8 are given by

E(k)(δ, x) =

{

η : η ∈ PCk−1([−a − cx, 0], [−δ, δ]), µx2 + σxη(0) > 0,∣

∣

∣

d j

dθ j η(θ)∣

∣

∣ 6 D j(δ) for θ ∈ [−a − cx, 0], j = 0, . . . , k − 1

}

. (3.6)

Let r+ = a + cδ then the auxiliary ODE problem (2.10) becomes

{

v(θ) = µv(θ) + ση(θ), θ ∈ [−r+, 0],

v(−r+) = η(0).(3.7)

Integrating (3.7) yields,

v(0) = η(0)eµr+ + σ

∫ 0

−r+

e−µθη(θ)dθ. (3.8)

Since the DDE (1.2) is scalar the set of x such that |x| = δ consists of just two points x = δ and

x = −δ. Suppose first that x = δ, then (3.6) implies that η(0) ∈ [−δ,−δµ/σ].

Definition 3.5. Let a > 0, c , 0, σ 6 µ and σ < −µ. For any δ ∈ (0, |a/c|) and u ∈ [−δ,−δµ/σ],

define r+ = a + cδ and η(k)(θ) for θ ∈ [−r+, 0] by

η(k)(θ) = infη∈E(k)(δ,δ)η(0)=u

η(θ). (3.9)

We also define the function I(u, δ, c, k) to be

I(u, δ, c, k) = ueµr+ + σ

∫ 0

−r+

e−µθη(k)(θ)dθ. (3.10)

The function η(k) given by (3.9) is the most negative one in E(k)(δ, δ) satisfying η(0) = u, and so

since σ < 0, this function maximizes v(0) for fixed η(0) by maximising the second term in (3.8).

This is the reason for considering η ∈ PCk−1([−a − cx, 0], [−δ, δ]) in the definition of E(k)(δ, x).

15

We really want to maximise v(0) for η ∈ E(k)(δ, x), where the smaller set E(k)(δ, x) is defined in

(2.11). Even though all the functions η ∈ E(k)(δ, x) satisfy η ∈ Ck−1([−a − cx, 0], [−δ, δ]), the

maximiser will in general only be piecewise Ck−1. With Definition 3.5 we can derive bounds on

the solution v(0) of the auxiliary ODE (3.7) for all η ∈ E(k)(δ, x) in both cases where x = ±δ.

Lemma 3.6. Let a > 0, c , 0, σ 6 µ andσ < −µ. Let δ ∈ (0, |a/c|). The solution of the auxiliary

ODE system (3.7) satisfies

v(0) 6 supu∈[−δ,− µσ δ]

I(u, δ, c, k), ∀η ∈ E(k)(δ, δ), v(0) > − supu∈[−δ,− µσ δ]

I(u, δ,−c, k), ∀η ∈ E(k)(δ,−δ).

Proof. First consider x = δ. The function I(u, δ, c, k) comes from (3.8) and depends on c and

δ through r+. Since, as noted above, the choice of η(k) maximizes (3.10) for fixed u, the first

inequality in the statement of the lemma follows.

Next consider x = −δ, then (3.6) implies that η(0) ∈ [δµ/σ, δ]. This time we should consider

the most positive function in E(k)(δ,−δ) satisfying η(0) = u ∈ [δµ/σ, δ], to obtain a lower bound

on v(0) for all η ∈ E(k)(δ,−δ). However, the model DDE (1.2) is invariant under the transforma-

tion (u, c) 7→ (−u,−c), so this function is −η(k)(θ) and the second inequality follows.

Notice from (3.10) that the functions I(u, δ, c, k) and I(u, δ,−c, k) only differ in their inte-

gration limits with I(u, δ, c, k) integrating η(k) over the interval [−a − cδ, 0] and I(u, δ,−c, k)

integrating over [−a + cδ, 0]. The integration over the larger of these intervals will be important

in the following sections and so it is convenient to define

P(δ, c, k) = supu∈[−δ,−δµ/σ]

I(u, δ, |c|, k). (3.11)

Comparing the cases when x = δ and −δ has to be done separately for each value of k, and we

can also explicitly define the functions η(k) for each k. This is handled in the following section

where we show that P(δ, c, k) < δ implies F (δ) < δ, and apply Theorem 2.7 to obtain asymptotic

stability for{

(µ, σ) : P(δ, c, k) < δ}

.

Barnea [1] applied Lyapunov-Razumikhin techniques to the c = 0 constant delay case of the

model DDE (1.2). His results do not apply to state-dependent case, as they were based on a result

for autonomous RFDEs which assumed F was Lipschitz, and he did not define an auxiliary ODE,

nor sets similar to E(k)(δ, x) or E(k)(δ, x). However, he did define functions η(k) for the constant

delay case by considering the most negative bounded functions with k − 1 bounded derivatives

as the function segments in the RFDE. In the limit as c → 0 our η(k) functions reduce to those

found by Barnea for the constant delay case. Because of the linearity of (1.2) with c = 0, Barnea

did not have to consider the upper and lower bounds separately as we did in Lemma 3.6, but

did define a function which is equivalent to P(δ, 0, k) in (3.11). Our asymptotic stability results

for the state-dependent model DDE (1.2) constitute a significant generalisation of the Lyapunov

stability results of Barnea [1] for the constant delay case, and moreover in Section 5 we will

correct an error of Barnea for the k = 2 constant delay case.

4. Asymptotic stability for u(t) = µu(t) + σu(t − a − cu(t)) using E(k)(δ, x)

In this section we consider the model state-dependent DDE (1.2) and use Theorem 2.7 to show

that the steady state is asymptotically stable in various parameter sets. In Theorem 4.1 we use

the set E(1)(δ, x) to show that the steady state is asymptotically stable whenever the parameters

16

values (µ, σ) are in the cone Σ∆. In the rest of the section we consider parameters in the wedge

and the cusp (Σw ∪Σc), and use Theorem 2.7 with k = 1, 2 and 3 to show the steady state is

asymptotically stable for (µ, σ) ∈ {

P(1, 0, k) < 1}

, where P(δ, c, k) is defined in (3.11). The sets{

P(1, 0, k) < 1}

are nested in Σw ∪Σc, becoming larger with k. We also find lower bounds on the

basin of attraction of the steady state. For the constant delay case (c = 0) the parameter regions

found in Σw ∪Σc are independent of the choice of δ in E(k)(δ, x). For the state-dependent case,

these regions change with c and δ (see Figure 4) and converge to the region for the constant delay

case as δ→ 0.

We begin by showing asymptotic stability in the cone Σ∆. The following result could also be

shown by adapting a stability result for time-dependent delays, such as that of Yorke [43]. Recall

that M0 is defined by (3.1).

Theorem 4.1 (Asymptotic stability for (1.2) in Σ∆). Let a > 0, c , 0 and |σ| < −µ so (µ, σ) ∈ Σ∆.

If |ϕ(t)| 6 |M0| for t ∈ [−a − c|M0|, 0] then the solution u(t) to (1.2) satisfies u(t)→ 0 as t → ∞.

Proof. With |x| = δ and µ < −|σ| it is impossible to satisfy µx2 + σxη(0) > 0 with |η(0)| 6 δand so E(1)(δ, x) is empty and asymptotic stability of the steady state follows from Theorem 2.7.

This holds for all δ ∈ (0, |M0|] and it follows directly from Theorem 2.7 that u(t) → 0 as t → ∞provided |ϕ(t)| < |M0|e−Lr(|M0 |) for t ∈ [−a − c|M0|, 0]. However, the exponential correction term

e−Lr(|M0 |) comes from using Lemma 2.3 in the proof of Theorem 2.7 to ensure that |u(t)| < |M0|for t ∈ [0, r(|M0|)]. But Lemma 3.2 already ensures that |u(t)| < |M0| for all t > 0 if |ϕ(t)| 6 |M0|for the model DDE (1.2); the result follows.

We already noted in Section 3 that Assumption 2.1 is satisfied for (1.2), and derived an ex-

pression for E(k)(δ, x) and indicated which are the most relevant functions in these sets. For k = 1

we do not need any boundsD j(δ) and the members of the set E(1)(δ, x) need not be continuous.

Then the function η(1) from (3.9) is given by,

η(1)(θ) =

{

−δ, θ ∈ [−a − cδ, 0),

u, θ = 0.(4.1)

For k = 2 we need to find D1(δ) such that | ddθη(θ)| 6 D1(δ) for all η ∈ E(2)(δ, x), where

E(2)(δ, x) is defined by (3.5). For η ∈ E(2)(δ, x) we have

η(θ) = u(2r + θ − a − cu(2r + θ)) for θ ∈ [−a − cx, 0] ⊆ [−r, 0],

where |u(t)| 6 δ for t ∈ [−r, 2r] and solves (1.2) for r > 0. We easily derive that |u(t)| 6|µu(t)| + |σu(t − a − cu(t))| 6 (|µ| + |σ|)δ for t ∈ [0, 2r]. Then

η′(θ) = ddθ

u(

2r + θ − a − cu(2r + θ))

=(

1 − cu(2r + θ))

u(

2r + θ − a − cu(2r + θ))

.

Hence |η′(θ)| 6 D1δ for θ ∈ [−r, 0] where

D1 =(|µ| + |σ|)(1 + (|µ| + |σ|)|c|δ). (4.2)

Thus we can choose D1(δ) = D1δ (note that D1 also depends on δ) to define E(2)(δ, x) and we

obtain that E(2)(δ, x) ⊆ E(2)(δ, x). The function η(2) is given by (3.9), as

η(2)(θ) =

u + D1δθ, θ ∈[− δ+u

D1δ, 0

]

,

−δ, θ ∈ [−a − cδ,− δ+uD1δ

]

, when δ+uD1δ< r+ = a + cδ.

(4.3)

17

where D1 is defined by (4.2).

For η ∈ E(3)(δ, x) ⊆ E(2)(δ, x), the same bound on the first derivative of η applies, and we also

bound the second derivative as follows. We have

η(θ) = u(3r + θ − a − cu(3r + θ)) for θ ∈ [−a − cx, 0] ⊆ [−r, 0],

where |u(t)| 6 δ for t ∈ [−r, 3r] and solves (1.2) for r > 0. As above we have that |u(t)| 6(|µ| + |σ|)δ for t ∈ [0, 3r]. Now noting that t − a − cu(t) ∈ [0, 2r] for t ∈ [r, 3r] we have

|u(t)| =∣

∣

∣µu(t) + σu(t − a − cu(t)(1 − cu(t))∣

∣

∣ 6 (|µ| + |σ|)2(1 + |σc|δ)δ,

for t ∈ [r, 3r]. Then, since 3r + θ − a − cu(3r + θ) ∈ [r, 2r] for θ ∈ [−r, 0] it follows that

|η′′(θ)| =∣

∣

∣

∣

d2

dθ2u(

3r + θ − a − cu(3r + θ))

∣

∣

∣

∣

=

∣

∣

∣

∣

(

1 − cu(3r + θ))2

u(

3r + θ − a − cu(3r + θ)) − cu(3r + θ)

)

u(

3r + θ − a − cu(3r + θ))

∣

∣

∣

∣

6(

D21 + (|µ| + |σ|)3|c|δ)(1 + |σc|δ)δ = D2δ.

Hence for all η ∈ E(3)(δ, x) we have |η′′(θ)| 6 D2(δ) = D2δ where

D2 =(

D21 + (|µ| + |σ|)3|c|δ)(1 + |σc|δ), (4.4)

and limδ→0 D2 = (limδ→0 D1)2 = (|µ| + |σ|)2. Taking D1 and D2 to satisfy (4.2) and (4.4) ensures

that E(3)(δ, x) ⊆ E(3)(δ, x). Then the η(k) function from (3.9) for k = 3 can be defined by

η(3)(θ) = η(3)(θ + θshift), θ ∈ [−r+, 0] (4.5)

where

η(3)(θ) =

−δ, θ 6 0,

−δ + δ2D2θ

2, θ ∈ (0, D1

D2),

−δ − δD21

2D2+ δD1θ, θ >

D1

D2.

θshift =

(

2(u + δ)

D2δ

)12

, u ∈ [−δ,−δ + δD21

2D2],

u + δ +δD2

1

2D2

D1δ, u > −δ + δD

21

2D2.

(4.6)

Here θshift is a convenient device which allows us to define η(3)(θ) for all values of u by the single

function η(3)(θ) with the shift used to obtain the correct value of u.

The η(k) functions define I(u, δ, c, k) via equation (3.10) and P(δ, c, k) through equation (3.11).

For k = 1, using (4.1) we easily evaluate

P(δ, c, 1) = I(

−µδ/σ, δ, |c|, 1)

=

− µσδeµr + δ

σ

µ(1 − eµr), µ , 0,

−δσr, µ = 0.(4.7)

For k = 2, from (3.10) and (4.3), if δ+uD1δ

> r+ then

I(u, δ, c, 2) = ueµr+ + σ

∫ 0

−r+

e−µθ(u + D1δθ)dθ (4.8)

=

u[

eµr+ + σµ

(eµr+ − 1)]

+ σµ

D1δ[

1µ(eµr+ − 1) − r+eµr+

]

, µ , 0,

u + σr+u − σD1r2+

2δ, µ = 0,

(4.9)

18

−1 −0.75 −0.5 −0.25 0

−1

−0.75

−0.5

−0.25

0

k = 2

k = 3

(a) µ = 1, σ = −1.2, u = 0

−1 −0.75 −0.5 −0.25 0

−1

−0.75

−0.5

−0.25

0

k = 2

k = 3

(b) µ = 0.4, σ = −0.6, u = 0

η(θ)η(θ)

θθ

u u

Figure 3: Sample η(k)(θ) functions for a = 1, c = 0 and δ = 1. The η(k)(θ) functions are defined in (4.3) and (4.5).

while if δ+uD1δ< r+ then we have to split the integral into two parts and

I(u, δ, c, 2) = ueµr+ + σ

∫ − δ+uD1δ

−r+

e−µθ(−δ)dθ + σ∫ 0

− δ+uD1δ

e−µθ(u + D1δθ)dθ (4.10)

=

u(

eµr+ − σµ

)

+ σµδ[

D1

µ

(

eµδ+uD1δ − 1

)

− eµr+]

, µ , 0,

u − σδr+ + σ2D1δ

(δ + u)2, µ = 0.(4.11)

To determine P(δ, c, 2) we perform the integration in I(u, δ, |c|, 2) and find u to maximise this

function. If δ+uD1δ

> r then u ∈ [

(rD1 − 1)δ,−δµ/σ]. This is only possible in the region where

rD1 − 1 6 −µ/σ. From (4.9) we have

I(u, δ, |c|, 2) = I1(u, δ) :=

u[

eµr + σµ

(eµr − 1)]

+ σµ

D1δ[

1µ(eµr − 1) − reµr

]

, µ , 0,

u + σru − σD1r2

2δ, µ = 0.

(4.12)

If δ+uD1δ< r then u has another upper bound u < (rD1 − 1)δ so u ∈ [−δ,min

{

(rD1 − 1)δ,− µσδ}]

.

Since the integration is broken down into two parts in this case we label the expression we derive

as I2, and from (4.11) we have

I(u, δ, |c|, 2) = I2(u, δ) :=

u(

eµr − σµ

)

+ σµδ[

D1

µ

(

eµδ+uD1δ − 1

)

− eµr]

, µ , 0,

u − σδr + σ2D1δ

(δ + u)2, µ = 0.(4.13)

The main differences between the expressions for I(u, δ, c, 2) and I(u, δ, |c|, 2) are that the former

involve r+ = a+ cδ, and the latter use r = a+ |c|δ as well as being subject to different restrictions

on the values of u for which they apply. In (4.9),(4.11),(4.12) and (4.13) the µ = 0 expressions

equal the µ → 0 limit of the µ , 0 expressions. Results for µ = 0 thus follow from those for

µ , 0, and so we do not treat these cases separately below.

Theorem 4.2. Let a > 0, c , 0, σ 6 µ and σ < −µ. Let δ ∈ (0, |a/c|). If P(δ, c, 2) < δ then

P(δ, c, 2) =

{

I1(−δµ/σ, δ), if rD1 − 1 6 −µ/σ,I2(−δµ/σ, δ), if rD1 − 1 > −µ/σ, (4.14)

19

where I1 is defined by (4.12) and I2 is defined by (4.13).

Proof. See Appendix A.

We will not state an explicit expression for P(δ, c, 3). When needed, this can determined by

evaluating (3.11) numerically for k = 3.

We now prove four lemmas which will be needed for the proof of Theorem 4.7 where we show

asymptotic stability in the set{

(µ, σ) : P(1, 0, k) < 1}

.

Lemma 4.3. Let a > 0, c , 0, σ 6 µ and σ < −µ. Let δ ∈ (

0, |a/c|) and u ∈ [−δ,−δµ/σ] be

fixed. Then I(u, δ, c, 1) decreases with decreasing r+.

Proof. For r+ > 0,

∂

∂r+

(

ueµr+ + σ

∫ 0

−r+

e−µθη(1)(θ)dθ

)

= eµr+[

µu + ση(1)(−r+)]

= eµr+[

µu − σδ] > 0,

since σ < −µ and u ∈ [−δ,−δµ/σ].

Lemma 4.4. For k = 2 or 3, let a > 0, c , 0, σ 6 µ and σ < −µ. Let δ ∈ (0, |a/c|) and

u ∈ [−δ,−µδ/σ] be fixed. Let I(r+) be the expression for I(u, δ, c, k) as a function of only r+;

I(r+) = ueµr+ + σ

∫ 0

−r+

e−µθη(k)(θ)dθ,∂∂r+I(r+) = eµr+

[

µu + ση(k)(−r+)]

. (4.15)

(A) If µ 6 0, then ∂∂r+I(r+) > 0.

(B) If µ > 0 and η(k)(−r+) 6 0, then ∂∂r+I(r+) > 0.

(C) If ∂∂r+I(r+) 6 0, then µ > 0, η(k)(−r+) > 0, and I(u, δ, c, k) < δ.

Proof. Parts (A) and (B) are easy to show. Let ∂∂r+I(r+) 6 0. From the first two cases, this is

only possible if µ > 0 and η(k)(−r+) > 0.

Consider k = 2 first. Since η(2)(−r+) , −δ we are in the case δ+uD1δ> r+. Thus I(u, δ, c, 2) is

given by (4.9). Since η(2)(−r+) = u − D1δr+ > 0 implies −σµ

D1δr+ <σµ

u, we deduce

I(u, δ, c, 2) 6 u[

eµr+ +σ

µ(eµr+ − 1)

]

+ δσD1

µ2(eµr+ − 1) +

σ

µueµr+ ,

= −uσ

µ+

(

1 + 2σ

µ

)

ueµr+ + δσD1

µ2(eµr+ − 1),

6 −uσ

µ+σ

µueµr+ + δ

σD1

µ2(eµr+ − 1), since

σ

µ< −1 and u > η(2)(−r+) > 0,

=σ

µ

(

eµr+ − 1)(

u + D1

µδ)

< 0 < δ.

For k = 3, from D21/D2 6 1 it follows that

η(3)(D1/D2) = −δ +δD2

1

2D2

6 −δ + δ2= −δ

2< 0.

Since η(3) is an increasing function and we require η(3)(−r+) > 0, then −r+ + θshift >D1

D2. Thus

θ + θshift > D1/D2 for all θ ∈ [−r+, 0]. By the definition of η(3), in this case η(3)(θ) = η(2)(θ) for

θ ∈ [−r+, 0] and I(u, δ, c, 3) = I(u, δ, c, 2) < δ.

20

Lemma 4.5. For k = 1, 2 or 3, let a > 0, c , 0, σ 6 µ and σ < −µ. Let δ ∈ (

0, |a/c|). Then

(µ, σ) : P(δ, c, k) = supu∈[−δ,−δµ/σ]

I(u, δ, |c|, k) < δ

⊆

(µ, σ) : supu∈[−δ,−δµ/σ]

I(u, δ,−|c|, k) < δ

.

Proof. Recall from Section 3 that changing the sign of c in I(u, δ, c, k) only changes the value of

r+ = a + cδ. For k = 1 the result follows from Lemma 4.3.

For k = 2 or 3, let (µ, σ) ∈ {

P(δ, c, k) < δ}

and u ∈ [−δ,−δµ/σ]. Recall that σ < 0, while

η(k)(θ) is a nondecreasing function in θ. There are two cases to consider:

(i) If µu + ση(k)(−(a − |c|δ)) 6 0 then by Lemma 4.4(C), I(u, δ,−|c|, k) < δ.

(ii) If µu + ση(k)(−(a + |c|δ)) > µu + ση(k)(−(a − |c|δ)) > 0, then µu + ση(k)(−τ)) > 0 for all

τ ∈ (a − |c|δ, a + |c|δ). By equation (4.15), the expression for I(r+) is increasing over this

interval and thus, I(u, δ,−|c|, k) 6 I(u, δ, |c|, k) 6 P(δ, c, k) < δ.

Thus I(u, δ,−|c|, k) < δ and the result follows.

Lemma 4.6. For k = 1, 2 or 3, let a > 0, c , 0, σ 6 µ and σ < −µ. If 0 < δ∗ 6 δ∗∗ < |a/c| then

{

(µ, σ) : P(δ∗∗, c, k) < δ∗∗} ⊆ {

(µ, σ) : P(δ∗, c, k) < δ∗}

.

Proof. Increasing δ increases r = a+ |c|δ which is the only source of nonlinearity in δ in the first

expression (4.7) for P(δ, c, 1). Thus for µ , 0

∂

∂δ

(

P(δ, c, 1)

δ

)

= −eµr(

µ

σ+σ

µ

)

∂

∂δ(µr) = µ|c|

(

µ2 + σ2

−σµ

)

eµr > 0. (4.16)

Positivity also follows trivially from (4.7) when µ = 0. The result follows for k = 1.

For k = 2 or 3, consider I(sδ, δ, |c|, k)/δ and note that r, D1, . . . ,Dk−1 are the only terms in the

expression that depend on δ, and that increasing δ increases r, D1 and D2. Thus

∂

∂δ

(

I(sδ, δ, |c|, k)

δ

)

=∂

∂r

(

I(sδ, δ, |c|, k)

δ

)

|c| +k−1∑

j=1

∂

∂D j

(

I(sδ, δ, |c|, k)

δ

)

∂D j

∂δ.

We focus on the first term on the left-hand side, since all the remaining terms are positive. From

(3.10) we can write

∂

∂r

(

I(sδ, δ, |c|, k)

δ

)

= eµr[

µs + σδη(k)(−r)

]

= eµr[

µs + σδη(k)(−(a + |c|δ)

]

.

Let r∗ = a + |c|δ∗, r∗∗ = a + |c|δ∗∗ and (µ, σ) ∈ {

P(δ∗∗, c, k) < δ∗∗}

. Let s ∈ [−1,−µ/σ] and

use the notation η(k)(δ, θ) to denote the function η(k) as a function of both θ and δ. Note that

η(k)(δ,−(a + |c|δ))/δ is always decreasing with δ. Consider the following cases:

(i) If µs + ση(k)(δ∗,−r∗)/δ∗ 6 0 then by Lemma 4.4(C), I(sδ∗, δ∗, |c|, k) < δ∗.

(ii) If µs + ση(k)(δ∗∗,−r∗∗)/δ∗∗ > µs + ση(k)(δ∗,−r∗)/δ∗ > 0 then ∂∂r

(I(sδ,δ,c,k)

δ

)

> 0 for δ ∈[δ∗, δ∗∗]. Thus, ∂

∂δ

(I(sδ,δ,c,k)

δ

)

> 0 for δ ∈ [δ∗, δ∗∗] and,

I(sδ∗, δ∗, c, k)

δ∗6I(sδ∗∗, δ∗∗, c, k)

δ∗∗6

P(δ∗∗, c, k)

δ∗∗< 1.

21

−4 0 2−4

0

2

(a) {P(1, 0, 1) < 1}

σ

µ -4 0 2 -4

0

2

(b) {P(1, 0, 2) < 1}

σ

µ

Figure 4: For (a) k = 1 and (b) k = 2, the set {P(1, 0, k) < 1} is shaded green and the boundary of {P(1/2, 1, k) < 1/2}is drawn in red. If (µ,σ) ∈ {P(1, 0, k) < 1} then the zero solution to (1.2) is asymptotically stable by Theorem 4.7. As

δ→ 0 the proof of Theorem 4.7 shows that {P(δ, c, k) < δ} converges to {P(1, 0, k) < 1}.

Cases (i) and (ii) both yield I(sδ∗, δ∗, |c|, k) < δ∗. Since this holds for all s ∈ [−1,−µ/σ],

P(δ∗, |c|, k) < δ∗ follows.

With these lemmas we can prove our main result.

Theorem 4.7 (Asymptotic stability for (1.2) using E(k)(δ, x)). For k = 1, 2 or 3, let a > 0, c , 0,

and (µ, σ) ∈ {P(1, 0, k) < 1} where P(δ, c, k) is defined by (3.11). Then (µ, σ) ∈ {P(δ1, c, k) < δ1}for some δ1 ∈

(

0, |a/c|). Furthermore, for δ ∈ (0, δ1] let δ2 = δe−k(|µ|+|σ|)(a+|c|δ) and |ϕ(t)| < δ2 for

all t ∈ [−a−|c|δ, 0], then the solution to (1.2) satisfies |u(t)| 6 δ for all t > 0 and limt→∞ u(t) = 0.

Proof. For this proof define

J =⋃

δ∈(0,|a/c|)

{

P(δ, c, k) < δ}

.

First we show that J = {P(1, 0, 1) < 1}. When c = 0 it is seen that I(sδ, δ, 0, k)/δ is independent

of δ for k = 1, 2 or 3. From this it follows that P(δ, 0, k)/δ = P(1, 0, k). Moreover, for all c, when

δ → 0 then r → a, and I(sδ, δ, |c|, k)/δ → I(sδ, δ, 0, k)/δ, since c only appears multiplied by δ

in these expressions. Thus P(δ, c, k)/δ → P(1, 0, k) as δ → 0. Because of this and Lemma 4.6,

J = {P(1, 0, k) < 1}.Let (µ, σ) ∈ {P(1, 0, k) < 1}. The existence of δ1 such that (µ, σ) ∈ {P(δ1, c, k) < δ1} follows

from the above discussion. It also follows that (µ, σ) ∈ {P(δ, c, k) < δ} for all δ ∈ (0, δ1].

Let δ ∈ (0, δ1]. Consider the auxiliary ODE (3.7). For all η ∈ E(k)(δ, δ) it follows from

Lemmas 3.6 and 4.5 that v(0) 6 supu∈[−δ,−δµ/σ] I(u, δ, c, k) 6 P(δ, c, k) < δ. Similarly, for all

η ∈ E(k)(δ,−δ) we obtain v(0) > − supu∈[−δ,−δµ/σ] I(u, δ,−c, k) > −P(δ, c, k) > −δ. Thus (2.12)

holds for all η ∈ E(k)(δ, x) or any |x| = δ. This is true for all δ ∈ (0, δ1]. Since E(k)(δ, x) ⊆ E(k)(δ, x),

applying Theorem 2.7 completes the proof.

For given (µ, σ) the condition P(1, 0, k) < 1 ensures (2.12) is satisfied for all η ∈ E(k)(δ, x) and

hence Theorem 4.7 establishes asymptotic stability for (µ, σ) in the part of Σw ∪Σc for which

22

P(1, 0, k) < 1. These sets are shown in Figure 4 for k = 1 and k = 2. The stability region

{P(1, 0, 1) < 1} shown in Figure 4(a) for the DDE (1.2) comprises a relatively small part of

Σw ∪Σc, because it is derived by requiring that (2.12) holds for all η ∈ E(1)(δ, x). But E(1)(δ, x) is

a very large set, with the main restrictions on η being that it is merely piecewise continuous with

‖η‖ 6 δ.We obtain a larger stability region by increasing k. This is seen in Figure 4(b) where

P(1, 0, 2) < 1 ensures that (2.12) is satisfied for all η ∈ E(2)(δ, x) results in a significantly

larger stability region than seen in Figure 4(a). Since E(2)(δ, x) ⊂ E(1)(δ, x), with all functions

η ∈ E(2)(δ, x) satisfying the derivative bound |η′(θ)| 6 D1δ, the set E(2)(δ, x) is smaller than

E(1)(δ, x) and it is possible to satisfy (2.12) over a larger region of (µ, σ) parameter space. We

will compare the sizes of the stability regions {P(1, 0, k) < 1} for different k in Section 5.

The boundary of {P(1/2, 1, k) < 1/2} is also shown in Figure 4 for k = 1 and k = 2. As δ→ 0

the sets {P(δ, c, k) < δ} converge to the set {P(1, 0, k) < 1}, and for (µ, σ) ∈ {P(1, 0, k) < 1}the inequality P(δ, c, k) < δ can be used to determine the largest δ1 and hence the largest δ2 for

which Theorem 4.7 applies. This determines a ball which is contained in the basin of attraction

of the steady state, and in Section 6 we consider how the size of this lower bound on the basin of

attraction varies with k.

5. Comparison of the stability regions

In this section we compare the sets in which we can establish asymptotic stability of the steady

state of the state-dependent DDE (1.2) using Lyapunov-Razumikhin techniques. In Sections 4

we showed asymptotic stability for (µ, σ) ∈ Σ∆, and for (µ, σ) in the parts of the cusp Σc and

wedge Σw for which P(1, 0, k) < 1 for k = 1, 2, 3. Measurements of these sets and the exact

stability region Σ⋆ are presented in Tables 1–3, and they are illustrated in Figure 5.

To compute these stability regions, from Theorems 4.7 we need to compute P(1, 0, k) in the

limiting case c = 0, δ = 1. This was done in MATLAB [36]. For k = 1 and 2 we have exact

expressions for P(1, 0, k) given by (4.7) and (4.14). Noting that σ < 0 in Σw ∪Σc, from (4.7) we

find that (µ, σ) satisfies P(1, 0, 1) < 1 when

σ2

µ(1 − eµa) − σ − µeµa > 0. (5.1)

The boundary of {P(1, 0, 1) < 1} is defined by equality in (5.1).

For k = 3 the value of P(1, 0, 3) was calculated by maximizing the function I(u, 1, 0, k) over

u ∈ [−1,−µ/σ] using the MATLAB fminbnd function.

The boundary of {P(1, 0, k) < 1} is then found by fixing one of µ or σ and using the fzero

function to find the value of the other one which solves P(1, 0, k)−1 = 0 (except in the case k = 1

where for given µ, applying the quadratic formula to (5.1) determines σ). The largest value of µ

for each region (shown in Table 3) is then found by regarding the µ that solves P(1, 0, k) = 1 as

a function of σ and using fminsearch to find the σ that maximises µ. The boundary of the full

stability domain Σ⋆, found by linearization, is given by Definition 3.4.

Since by Theorem 4.7, at least for k 6 3, the Lyapunov-Razumikhin stability regions are given

by P(1, 0, k) < 1 irrespective of the value of c, we obtain the same regions in the constant c = 0

and variable c , 0 delay cases. This is consistent with the linearization theory of Gyori and

Hartung [12] who showed that Σ⋆ is the exponential stability region for both c = 0 and c , 0.

When µ = 0 the DDE (1.2) becomes

u(t) = σu(t − a − cu(t)) (5.2)

23

−5 −2 0 2−5

−2

0

2

(a) Σ∆ ∪{P(1, 0, 1) < 1}

−5 −2 0 2−5

−2

0

2

(b) Σ∆ ∪{P(1, 0, 2) < 1}

−5 −2 0 2−5

−2

0

2

(c) Σ∆ ∪{P(1, 0, 3) < 1}

σσ

µ

σ

µ µ

Figure 5: Stability regions found using Theorems 4.1 and 4.7 shaded in green. Parameter pairs (µ,σ) ∈ Σ⋆ which are

contained in the wedge Σw or cusp Σc but not in {P(1, 0, k)} are shaded dark or light grey respectively.

Region σ at µ = −5 σ at µ = −2 σ at µ = 0

{P(1, 0, 1) < 1} −5.0673855 −2.5578041 −1

{P(1, 0, 2) < 1} −5.0676090 −2.5875409 −1.5 = −3/2

{P(1, 0, 3) < 1} −5.0678325 −2.6127906 −1.5416667 = −37/24

Σ⋆ −5.6605586 −3.0396051 −1.5707963 = −π/2

Table 1: Boundaries of the stability regions: Values of σ for fixed µ with a = 1.

and the intervals of σ values in the stability regions, (shown in the last column of Table 1) can

be found exactly. From (4.7) we have P(1, 0, 1) = −σa when µ = 0 which implies σ ∈ (−1/a, 0)

for P(1, 0, 1) < 1. Similarly, when σ < −1/a from (4.14) we have P(1, 0, 2) = −σa − 1/2 and

hence σ = −3/(2a) on the boundary of {P(1, 0, 2) < 1}. Magpantay [30] shows that for µ = 0 we

require σ ∈ (−37/(24a), 0) for P(1, 0, 3) < 1. For the constant delay case of (5.2), with c = 0,

Barnea [1] showed Lyapunov stability for σ ∈ [−3/(2a), 0], by applying Lyapunov-Razumikhin

techniques with k = 2. The stability bound σ > −37/(24a) seems not to have been derived

before for the constant delay case of (5.2), but is well-known for Wright’s equation [42] which is

a nonlinear constant delay DDE whose linear part corresponds to (5.2) with c = 0 (see [28, 42]).

When µ = 0, the boundary of {P(1, 0, k) < 1} seems to converge rapidly to −π/(2a), the

boundary of Σ⋆, as k → ∞, suggesting that the full stability interval can be recovered. Indeed,

for constant delay with µ = c = 0 Krisztin [26] showed Lyapunov stability for σ ∈ (−π/(2a), 0]

by considering k → ∞.

Barnea [1] and Myshkis [37] also applied Razumikhin techniques to establish Lyapunov sta-

bility for (1.2) in the case of constant delay (c = 0) with µ , 0. The regions in which they

claim stability are shown in Figure 6. The region found by Myshkis [37] has σ > −1/a and

µ 6 σ(aσ + 1)/(aσ − 1), which for a = 1 always has µ 6 3 − 2√

2 and is contained in

{P(1, 0, 1) < 1}.Barnea [1] claimed that the Lyapunov stability region of (1.2) with c = 0 contains the region

X2 ={

(µ, σ) : 0 6 s∗ 6 a, P < 1}

where

s∗ = −eµa

σ, P =

σ(µ + σ)

µ2

[

eµs∗ − σ

µ + σ

]

.

24

Region µ at σ = −5 µ at σ = −2 µ at σ = −1

{P(1, 0, 1) < 1} −4.9286634 −1.16401463 0

{P(1, 0, 2) < 1} −4.9283941 −1.0040856 0.38774807

{P(1, 0, 3) < 1} −4.9281247 −0.93607885 0.39925645

Σ⋆ −4.2734224 −0.63804505 1

Table 2: Boundaries of the stability regions: Values of µ for fixed σ with a = 1.

Region Supremum of µ Corresponding value of σ

{P(1, 0, 1) < 1} 0.18822641 = ln((1 +√

2)/2)) −0.45439453

{P(1, 0, 2) < 1} 0.45697166 −0.73935547

{P(1, 0, 3) < 1} 0.45700462 −0.74059482

Σ⋆ 1 −1

Table 3: The values of µ and σ at the rightmost boundary point of each stability region with a = 1.

We show this region in Figure 6(a), but Barnea did not actually graph X2 or give its derivation

in [1]. He noted that setting P = 1 and letting µ → 0 yields that the point σ = −3/2a is a

boundary of X2 on the σ-axis. We observe that setting s∗ = 0 and µ → 0 yields σ = −1/a as

the other boundary of X2 on the σ-axis. Thus the region X2 does not include the whole interval

σ ∈ (−3/2a, 0] on the σ-axis which Barnea had proven to be Lyapunov stable in the µ = 0

case in the same paper [1]. Barnea’s stability region X2 is hence incomplete. Although the η(2)

function used by Barnea to show Lyapunov stability corresponds to (4.3) with c = 0, it appears

that Barnea performed his integration assuming that δ+uD1δ

6 r+ in all cases. The case whenδ+uD1δ> r+ occurs in the µ = c = 0 case (as well as the general case c , 0, µ , 0 considered in

(4.12),(4.13)). Omitting this case results in the incorrect stability region X2. The correct region

is (µ, σ) ∈ {

P(1, 0, 2) < 1}

as illustrated in Figure 5(b). Moreover within this region we show the

stronger property of asymptotic stability for both the constant delay (c = 0) and state-dependent

delay (c , 0) cases.

Tables 1 and 2 also show that for µ < 0 we can show asymptotic stability in a larger part

of Σ⋆ by increasing k. However when µ ≪ 0 the improvement in going from k = 1 to 2 to 3

is very marginal and we can only show asymptotic stability in a slice of the wedge Σw whose

width appears to go to zero as µ → −∞. The problem here is that as µ → −∞ the DDE (1.2) is

singularly perturbed and can be written as the so-called saw-tooth equation

εu(t) = u(t) + Ku(t − a − cu(t))

where ε = 1/µ and K = σ/µ. This DDE had been studied in detail in [35] and for K > 1

sufficiently large (corresponding to (µ, σ) outside Σ⋆) the steady state is unstable, but there is

an asymptotically stable slowly oscillating periodic solution. This periodic solution, known as

the sawtooth solution, has unbounded gradient and a discontinuous profile in the singular limit.

For parameter values inside the wedge Σw the steady state is asymptotically stable, and for large

and negative µ there are no periodic solutions but a slowly decaying sawtooth-like oscillation

can occur. Lyapunov-Razumikhin techniques based on bounding derivatives of solutions cannot

perform well when those derivatives can be arbitrarily large. To improve the results in this case it

would be necessary to define different sets E∗(k)

(δ, x) which take into account the structure of the

oscillations and are hopefully much closer to E(k)(δ, x) than the sets E(k)(δ, x) that we use here.

25

-4 0 2 -4

0

2

(a)

σ

µ -4 0 2 -4

0

2

(b)

σ

µ

Figure 6: (a) The set X2 \ Σ∆ which is part of the stability region of (1.2) for the constant delay case (c = 0) according to

Barnea [1]. (b) The part of the stability domain outside Σ∆ for the same problem found by Myshkis [37].

For µ > 0 there is a significant improvement in the computed stability domain in going from

k = 1 to k = 2 and a smaller improvement using k = 3. The largest value of µ which satisfies

P(1, 0, 1) 6 1 can be computed from (5.1) which is quadratic in σ. Then non-negativity of the

discriminant imposes the bound that µ < (1/a) ln((1 +√

2)/2) ≈ 0.1882/a, as seen in Table 3.

Although the parameter regions in which we can show asymptotic stability are independent of

c, we will see in Section 6 that the basins of attraction do depend on c.

6. Basins of attraction

Theorem 4.7 shows that for (µ, σ) ∈ Σw ∪Σc the ball

{

ϕ : ‖ϕ‖ < δ2 = δ1e−k(|µ|+|σ|)(a+|c|δ1)}

(6.1)

is contained in the basin of attraction of the steady-state of the state-dependent DDE (1.2) for

k = 1, 2 and 3. For fixed δ1 the radius of this ball gets smaller as k increases, but the value of

δ1 depends on k, µ and σ, and some work is required to determine the largest such ball that is

contained in the basin of attraction. In [31] we show that (6.1) can be improved when µ < 0, so

here we will consider (µ, σ) ∈ Σc, where σ < 0 6 µ. Lemma 3.2 does not apply when µ > 0,

so there is no a priori bound on the solutions to (1.2) in this case. We present two examples

which show that (1.2) can have unbounded solutions when µ > 0, which also shows that the

steady-state is not globally asymptotically stable when (µ, σ) ∈ Σc and gives an upper bound on

the largest ball contained in its basin of attraction. For simplicity of exposition we suppose c > 0

in this section, but the results can easily be extended to c < 0. We first consider µ = 0.

Example 6.1. Consider (1.2) with c > 0, a > 0, µ = 0 and σ ∈ [−π/2a,−1/a) and for δ ∈[−1/(cσ), a/c) let ϕ(t) be Lipschitz continuous with

ϕ(0) = δ, ϕ(t) = −δ for t 6 −a − cδ,

ϕ(t) ∈ (−δ, δ) for t ∈ (−a − cδ, 0).

26

−1.5 −1.25 −1 −0.75 −0.5 −0.25 00

0.2

0.4

0.6

0.8

1

k=1k=2k=3δ*

(a) δ1 when µ = 0, σ ∈ [−π/2, 0]

σ

δ

−1.5 −1.25 −1 −0.75 −0.5 −0.25 00

0.2

0.4

0.6

0.8

1

k=1

k=2

k=3

δ*

(b) δ2 when µ = 0, σ ∈ [−π/2, 0]

σ

δ

Figure 7: For fixed a = c = 1, µ = 0 and σ ∈ [−π/2, 0], (a) supremum of δ1 ∈ (0, a/c] such that (µ, σ) ∈ {P(δ1, c, k) < δ1}with k = 1, 2, 3, and, (b) δ2 = δ1e−k|σ|(a+|c|δ1). The value of δ∗ from Example 6.1 is also shown in both plots.

Then while the deviated argument α(t, u(t)) = t − a − cu(t) 6 −a − cδ we have

u(t − a − cu(t)) = −δ and u(t) = −σδ.

Hence,

u(t) = δ(1 − σt) > δ +t

c, for t > 0. (6.2)

But now α(t, u(t)) 6 −a − cδ for all t > 0 and (6.2) is valid for all t > 0. Thus for µ = 0, σ ∈[−π/2a,−1/a) on the axis between the third and fourth quadrants of the stability region we have

‖ϕ‖ = δ and |u(t)| → ∞ as t → ∞. It follows that the steady state is not globally asymptotically

stable and also that B(0, δ) is not contained in its basin of attraction. Thus δ∗ = −1/cσ provides

an upper bound on the radius of the largest ball contained in the basin of attraction.

Figure 7 shows three bounds on the basin of attraction of the steady state of (1.2). The value

of δ∗ from Example 6.1 gives an upper bound on the radius of the largest ball contained in the

basin of attraction. Two lower bounds on the radius of the largest ball are also shown. The larger

bound δ1 gives the radius of the ball that Theorem 4.7 shows is contracted asymptotically to the

steady state provided the solution is sufficiently differentiable. Lemma 2.3 is used to ensure that

the solution remains bounded long enough to acquire sufficient regularity, and the growth in the

solution allowed by that lemma results in the smaller radius δ2 (as defined by (6.1)) of the ball

that is contained in the basin of attraction for general continuous initial functions ϕ. We see that

the bounds δ1 increase monotonically with k, but because of the exponential term in (6.1), the

largest value of δ2 is achieved with k = 1 in most of the interval for which P(1, 0, 1) < 1.

Now consider the case of µ > 0. We can again derive an upper bound on the basin of attraction

of the steady state when (µ, σ) ∈ Σc.

Example 6.2. Let a > 0, c > 0, µ > 0 and (µ, σ) ∈ Σc so σ < −µ < 0. Also let

q(δ) = −aµ − 1 − cσδ − ln(

cδ(µ − σ))

.

Note that q(1/(c(µ − σ))) = −µ(a + 1/(µ − σ)) < 0, while q′(δ) = −cσ − 1/δ < 0 for all

δ ∈ (0, 1/(c(µ − σ))). Also q(δ) → ∞ as δ → 0, hence there exists δ∗ ∈ (

0, 1/(c(µ − σ)))

such

that q(δ∗) = 0 and δ∗ is unique in this interval.

27

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

k=2k=3δ*

(a) δ1 when σ = −1, µ > 0

µ

δ

0 0.1 0.2 0.3 0.40

0.005

0.01

0.015

0.02

0.025

k=2k=3

(b) δ2 when σ = −1, µ > 0

µ

δ

Figure 8: For fixed a = c = 1, σ = −1 and µ > 0, (a) supremum of δ1 ∈ (0, a/c] such that (µ,σ) ∈ {P(δ1, c, k) < δ1} with

k = 2, 3, and, (b) δ2 = δ1e−k(|µ|+σ|)(a+|c|δ1). The value of δ∗ from Example 6.2 is also shown in (a).

Suppose that the parameters are chosen so that δ∗ < a/c. A sufficient (but not necessary)

condition for this is σ < µ − 1/a since this implies 1/(c(µ − σ)) 6 a/c. Now let δ ∈ (δ∗, a/c) so

q(δ) < 0 and consider (1.2) with ϕ(t) Lipschitz continuous and

ϕ(0) = δ, ϕ(t) = −δ for t 6 µq(δ),

ϕ(t) ∈ (−δ, δ) for t ∈ (µq(δ), 0),

Then (1.2) has solution

u(t) =σδ

µ+ δeµt

[

µ − σµ

]

(6.3)

with u(t − a − cu(t)) = −δ for all t > 0. To see this note that

α(t, u(t)) = t − a − cu(t) = t − a − cσδ

µ− cδeµt

[

µ − σµ

]

,

with α(t, u(t)) → −∞ as t → ∞. Differentiating the expression for α(t, u(t)) shows that

α(t, u(t)) 6 µq(δ) < 0 for all t > 0, with α(t, u(t)) = µq(δ) when t = −µ−1 ln (cδ(µ − σ)).

Hence, as in Example 6.1, we have ‖ϕ‖ = δ and |u(t)| → ∞ as t → ∞. The steady state is not

globally asymptotically stable and the ball B(0, δ) is not contained in its basin of attraction. Thus

δ∗ provides an upper bound on the radius of the largest ball contained in the basin of attraction.

For σ = −1 and µ > 0, Figure 8 shows the same bounds δ1, δ2 and δ∗ on the radius of the basin

of attraction of the steady state as were shown in Figure 7. Since these parameters are outside

the set {P(1, 0, 1) < 1} no bound is shown for k = 1. On nearly all of this interval k = 2 gives the

largest lower bound δ2 on the radius of a ball contained in the basin of attraction.

Figure 9 shows these bounds on the basin of attraction in the cusp Σc. The shaded region in

Figure 9(c) denotes the portion of Σc for which δ∗ 6 a/c when a = c = 1, and hence δ∗ from

Example 6.2 gives an upper bound on the radius of the largest ball contained in the basin of

attraction. The corresponding bounds δ∗ are shown as contours within this region. Figure 9(a)

and (b) shows the lower bounds δ1 and δ2, along with the value of k that achieves the bound.

In all three figures in this section, δ2 is computed using (6.1) and δ1 is obtained from solving

P(δ1, c, k) = δ1 similarly to computations described in Section 5.

28

0 0.25 0.5 0.75 1

−1.5

−1

−0.5

0

00.020.050.10.2

0.50.751

k=3

(a) δ1

µ

σ

0 0.25 0.5 0.75 1

−1.5

−1

−0.5

0

00.0010.0050.010.02

0.050.10.

2

0.50.75

k=3

k=2

k=1

(b) δ2

µ

σ

0.1

0.2

0.2

0.5

0.5

0.75

0.75

1

1

0 0.5 1

−1.5

−1

0

(c) δ∗µ

σ

Figure 9: Plot of Σc for fixed a = c = 1 with contour plots of (a) the maximum δ1 ∈ (0, a/c] such that (µ,σ) ∈{P(δ1, c, k) < δ1}, and (b) the δ2 that maximizes δ2 = δe

−k(|µ|+|σ|)(a+|c|δ for δ ∈ (0, δ1]. Shading shows the value k ∈ {1, 2, 3}for which the maximum is achieved. (c) The upper bound δ∗ from Example 6.2 for the radius of the largest ball B(0, δ)

contained in the basin of attraction of the zero solution to (1.2).

7. Conclusions

In this paper we have expanded upon the existing work on Lyapunov-Razumikhin techniques

by providing results specifically tailored to DDEs with time-varying discrete delays including

problems with state-dependent delays and vanishing delays. Our main results provide sufficient

conditions for Lyapunov and asymptotic stability of steady state solutions of DDEs in Theo-

rems 2.5 and 2.7 respectively. These conditions involve converting the DDE into a corresponding

ODE problem with the delay terms treated as source terms that satisfy constraints. Our results

require a Lipschitz condition on the right-hand side function f in (1.1) instead of the more restric-

tive Lipschitz condition on F in (1.4) required in Barnea [1], and do not require the construction

of auxiliary functions as required by Hale and Verduyn Lunel [15]. Nevertheless we are able

to show asymptotic stability, using a proof by contradiction showing that there cannot exist a

solution which is not asymptotic to the steady state.

We apply our results to the model state-dependent DDE (1.2) in Sections 4–6. The main result

of the application of Lyapunov-Razumikhin techniques to (1.2) is given as Theorems 4.7 where

29

we prove that the zero solution to (1.2) is asymptotically stable if (µ, σ) ∈ {P(1, 0, k) < 1}, for

k = 1, 2 or 3 and provide lower bounds on the basin of attraction.

The parameter regions in which stability is proven in these theorems are compared in Section 5.

As shown in Figure 5, the derived parameter regions grow as larger values of k are used, though

for µ , 0, the derived stability region does not approach the entire known stability region Σ⋆ as

k → ∞ (for reasons discussed in in Section 5).

In Section 6 we consider (1.2) in the cusp Σc where µ > 0 and the steady state would be

unstable without the delay term. In Examples 6.1 and 6.2 we constructed solutions which do not

converge to the steady state for (µ, σ) ∈ Σc. These solutions provide us with an upper bound δ∗

on the radius of the largest ball about the zero solution contained in the basin of attraction. In

Figures 7–9 these upper bounds were compared with the lower bound δ2 on the basin of attraction

from (6.1).

In the current work have studied stability through Lyapunov-Razumikhin techniques, but let

us briefly compare and contrast this approach to the alternative, namely linearization. State-

dependent DDEs have long been linearized by freezing the delays at their steady-state values and

linearizing the resulting constant delay DDE [4, 5]. This heuristic approach has recently been

put on a rigorous footing. For a class of state-dependent DDEs which includes (1.2) with µ = 0,

Gyori and Hartung [11] proved that the steady state of the state-dependent DDE is exponentially

stable if and only if the steady state of the corresponding frozen-delay DDE is exponentially

stable. In [12] they generalise this result to a class of nonautonomous problems which are linear

except for the state-dependency.

To compare and contrast our results with the linearization results of [12], we note that our

results apply to a larger class of problems (1.1) than was considered in [12], and we prove both

Lyapunov stability and asymptotic stability results, whereas [12] is concerned with exponen-

tial stability. The results in [12] do apply directly to our model problem (1.2), and reveal the

parameter region for which the steady state is exponentially stable. In contrast our Lyapunov-

Razumikhin techniques are only able to deduce stability in part of this parameter region.

Even though Lyapunov-Razumikhin techniques do not provide a proof of stability in the entire

known stability region for (1.2), just as Lyapunov functions for ODEs do not always do so, they

can nevertheless still be a very useful tool for studying stability in state-dependent DDEs. In

particular our Lyapunov stability result is applicable to nonautonomous problems (for some of

which rigorous linearization has yet to be derived) and the asymptotic stability result yield bounds

on the basins of attraction which cannot be derived through linearization.

Acknowledgments

ARH is grateful to Tibor Krisztin, John Mallet-Paret, Roger Nussbaum and Hans-Otto Walther

for productive discussions and suggestions, and to the National Science and Engineering Re-

search Council (NSERC), Canada for funding through the Discovery Grant program. FMGM

is grateful to Jianhong Wu for helpful discussions, and to McGill University, York University,

the Institut des Sciences Mathematiques (Montreal, Canada) and NSERC for funding. We are

grateful to an anonymous referee whose feedback significantly improved the manuscript.

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Appendix A. An explicit expression for the region P(δ, c, 2) < δ

Here we prove Theorem 4.2. Let I1 and I2 be defined by (4.12) and (4.13). Recall that I1 only

applies when δ+uD1δ

> r, in which case the integration does not have to be split into two intervals.

For this case to occur we require u ∈ [(rD1 − 1)δ,−µδ/σ], which is only possible in the region

where rD1 − 1 6 −µ/σ. When δ+uD1δ< r the integration has to be broken into two parts and I2

applies. In that case u ∈ [−δ,min{

(rD1 − 1)δ,−µδ/σ}]. We require the following lemmas.

Lemma A.1. Let a > 0, c , 0, σ 6 µ and σ < −µ. Let δ ∈ (0, |a/c|) and rD1 − 1 6 −µ/σ. If

µ > 0 then σ > −1/r. If µ < 0 then µ ∈ [

(−3 + 2√

2)/r, 0]

and

σ > − 1r

[

12(1 + µr) + 1

2

√

1 + 6µr + (µr)2

]

> − 1r. (A.1)

Proof. Let rD1 − 1 6 −µ/σ. Then

r(sign(µ)µ − σ) − 1 = r(|µ| + |σ|) − 1 6 rD1 − 1 6 − µσ,

⇒ rσ2 + (1 − sign(µ)µr)σ − µ 6 0. (A.2)

The boundary of the region where this inequality holds is

σ = − 1r

[

12(1 − sign(µ)µr) ± 1

2

√

(1 − sign(µ)µr)2 + 4µr]

. (A.3)

If µ > 0 then this simplifies to σ = −1/r and σ = µ. Since σ 6 0, and also µ = σ = 0 satisfies

(A.2) it follows that (A.1) holds for σ ∈ [−1/r, 0] for the case µ > 0.

If µ < 0 then (A.3) simplifies to

σ = − 1r

[

12(1 + µr) ± 1

2

√

1 + 6µr + (µr)2]

. (A.4)

Requiring 1+ 6µr+ (µr)2 > 0 yields µr ∈ [−3+ 2√

2, 0]. The lower bound on σ can be found by

taking the lower boundary in (A.4) which attains its minimum at µ = 0. This yields (A.1).

32

Lemma A.2. Let a > 0, σ 6 µ and σ < −µ. Let δ ∈ (0, |a/c|). Define

u∗ =[

1µ

D1 ln(

1 − µσ

eµr)

− 1]

δ.

The following statements are true:

(A) If rD1 − 1 > −µ/σ then the maximum of I2(u, δ) over u ∈ [−δ,−µδ/σ] occurs at u∗ if

u∗ < −µδ/σ, and at −µδ/σ otherwise.

(B) If rD1 − 1 > −µ/σ and u∗ < −µδ/σ then P(δ, c, 2) > δ.

(C) If rD1 − 1 6 −µ/σ then supu∈[−δ,(rD1−1)δ] I2(u, δ) = I2((rD1 − 1)δ, δ)

(D) If rD1 − 1 6 −µ/σ then supu∈[(rD1−1)δ,−µδ/σ] I1(u, δ) = I1(−µδ/σ, δ)(E) If rD1 − 1 6 −µ/σ then I2((rD1 − 1)δ, δ) 6 I1(−µδ/σ, δ).

Proof of (A). To find u∗ which maximises I2(u∗, δ) consider the derivative

∂∂uI2(u, δ) = eµr + σ

µ

(

eµδ+uD1δ − 1

)

. (A.5)

At u = −δ this is positive. Setting the derivative equal to zero in (A.5) yields

u∗ =[

1µ

D1 ln(

1 − µσ

eµr)

− 1]

δ.

Since 1 − µσ

eµr ∈ [0, 1] if µ < 0 and 1 − µσ

eµr > 1 if µ > 0 then u∗ > −δ in both cases.

Proof of (B). First we need to prove that if u∗ < −µδ/σ and µ > 0 then rD1 − 1 6 −µ/σ. Let

u∗ < −µδ/σ and µ > 0. Then ∂∂uI2(−µδ/σ, δ) < 0. Consider the term D1/µ,

D1

µ=|µ| + |σ|µ

(

1 + (|µ| + |σ|)|c|δ) > |µ| + |σ|µ

=

(

1 − σµ

)

. (A.6)

Now consider the exponent of the second term in (A.5) with u = −µδ/σ,

µ1 − µ/σ

D1

=

(

1 − µσ

)

µ

D1

61 − µ/σ1 − σ/µ = −

µ

σ.

Thus,

eµr + σµ

(e−µ/σ − 1) 6 ∂∂uI2(−µδ/σ, δ) < 0.

Isolating r in this expression yields r < 1µ

ln[σµ

(

1 − e−µ/σ)]

. Let x = µ/σ, then x ∈ (−1, 0) and

(1 − e−x)/x > 1. Also, (1 − 1/x) ln[(1 − e−x)/x] − 1 6 −x. These inequalities and (A.6) imply

rD1 − 1 6D1

µln

[

σµ

(1 − e−µ/σ)]

− 1 6(

1 − σµ

)

ln[

σµ

(1 − e−µ/σ)]

− 1 6 − µσ.

Thus if rD1 − 1 > −µ/σ then u∗ < −µδ/σ can only occur if µ < 0.

Now let µ < 0 and u∗ < −µδ/σ. Then by setting ∂∂uI2(u∗, δ) = 0 in (A.5), e

µδ+u∗D1δ − 1 = − µ

σeµr.

Also, eµr − σ/µ < 0 because µ < 0 and σ 6 µ < 0. Thus,

I2(u∗, δ) = u∗(

eµr − σµ

)

+ σµδ

[

D1

µ

(

eµδ+u∗D1δ − 1

)

− eµr]

,

> − µσδ(

eµr − σµ

)

+ σµδ[

−D1

σeµr − eµr

]

= δ −(

D1

µ+µ

σ+ σµ

)

δeµr.

But D1 > |µ| + |σ| > |µ/σ||µ| + |σ| = −(µ2/σ + σ) which impliesD1

µ+µ

σ+ σµ6 0. Hence

I2(u∗, δ) > δ as required.

33

−5 0 2−5

0

2

µ

σ

Figure 10: Region where rD1 − 1 6 −µ/σ is shown in brown and the region where eµr + (σ/µ)(eµr − 1) < 0 is shown in

blue. These two regions do not intersect.

Proof of (C). Let ∂∂uI2((rD1 − 1)δ, δ) < 0. Then, eµr + σ

µ(eµr − 1) < 0, which can be rewritten as

σ <µeµr

1 − eµr= −1

r

(

−µreµr

1 − eµr

)

. (A.7)

We show that the expression on the right-hand-side is continuous and decreases as µ increases.

Let x = µr, then

limµ→0−1

r

(

−µreµr

1 − eµr

)

= limx→0−1

r

(

−xex

1 − ex

)

= −1

r,

d

dµ

[

−1

r

(

−µreµr

1 − eµr

)]

=d

dx

xex

1 − ex=

ex (1 + x − ex)

(1 − ex)26 0.

So when µ > 0, a necessary condition for (A.7) to hold is σ < −1/r. From Lemma A.1, if

rD1 − 1 6 −µ/σ and µ > 0 then σ > −1/r. Thus ∂∂uI2((rD1 − 1)δ, δ) > 0 if rD1 − 1 6 −µ/σ

and µ > 0.

Now let rD1 − 1 6 −µ/σ and µ < 0. From Lemma A.1, µr ∈ [−3 + 2√

2, 0] and σ >

− 1r

[

12

(1 + µr) − 12

√

1 + 6µr + (µr)2]

. Since −xex

1−ex >12

(1 + x) + 12

√1 + 6x + x2 for x ∈ [−3 +

2√

2, 0], then σ > − 1r

(−µreµr

1−eµr

)

. This contradicts (A.7). Thus ∂∂uI2((rD1 −1)δ, δ) > 0 if rD1 −1 6

−µ/σ and µ < 0. Thus,

rD1 − 1 6 − µσ⇒ ∂

∂uI2((rD1 − 1)δ, δ) = eµr + σ

µ(eµr − 1) > 0. (A.8)

This is shown in Figure 10. To finish the proof of (C), observe from (A.5) that ∂∂uI2(u, δ) de-

creases as u increases. Then by (A.8), supu∈[−δ,(rD1−1)δ] I2(u, δ) = I2((rD1 − 1)δ, δ).

Proof of (D). Let rD1 − 1 6 −µ/σ. For all u ∈ [

(rD1 − 1)δ,−µδ/σ],∂∂uI1(u, δ) = eµr + σ

µ(eµr − 1) = ∂

∂uI2((rD1 − 1)δ, δ).

From (A.8), ∂∂uI1(u, δ) > 0 for all u ∈ [−δ, (rD1 − 1)δ]. Thus, supu∈[(rD1−1)δ,−µδ/σ] I1(u, δ) =

I1(−µδ/σ, δ).34

Proof of (E). From (4.12) and (4.13) we find that

I1(−µδ/σ, δ) − I2((rD1 − 1)δ, δ) =(

− µσ− (rD1 − 1)

)

δ[

eµr + σµ

(eµr − 1)]

.

For rD1 − 1 6 −µ/σ it follows from (A.8) that this expression is non-negative. Thus,

I1(−µδ/σ, δ) > I2((rD1 − 1)δ, δ).

Finally we can prove Theorem 4.2.

Proof of Theorem 4.2. Note that this expression does not hold outside of{

P(δ, c, 2) < δ}

. In order

to prove this theorem, we need items (A)-(E) in Lemma A.2 which require Lemma A.1.

Let rD1−1 > −µ/σ. Then we can only have the two-part integration so I(u, δ, c, 2) = I2(u, δ).

From (A) and (B), P(δ, c, 2) = I(−µ/σ, δ) if P(δ, c, 2) < δ.

Let rD1 − 1 6 −µ/σ. Then we can have either the one-part or the two-part integration.

From (C) and (D), I(u, δ, c, 2) = max{I2((rD1 − 1)δ, δ),I1(−µδ/σ, δ)}. From (E), P(δ, c, 2) =

I1(−µδ/σ, δ).

35